#72238166

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 24962 conjugacy classes of subgroups; only the 31771 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

(5,6,9,8)(7,10,11,12)(14,20)(18,28)(21,22)(23,27);(2,4)(6,8)(10,12)(13,14)(15,18)(16,21)(17,23)(19,20)(22,25)(24,27)(26,28);(5,6,9,8)(7,10,11,12)(13,15)(14,18)(16,24)(17,25)(19,26)(20,28)(21,27)(22,23);(2,4)(6,8)(10,12)(13,16,19,25)(14,21,20,22)(15,24,26,17)(18,27,28,23);(2,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,22)(15,24)(17,26)(18,23)(19,25)(20,21)(27,28);(5,8,9,6)(7,10,11,12)(13,17,19,24)(14,23,20,27)(15,25,26,16)(18,22,28,21);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,18,19,28)(14,15,20,26)(16,27,25,23)(17,21,24,22)

Polycyclic presentation:

2^10|a^2=h,c^2=h,d^2=i,f^2=hi,g^2=j,b^a=bhi,c^b=ch,d^a=dh,d^c=dh,e^b=ei,e^d=ei,f^b=fh,f^d=fh,f^e=fhi,g^a=gi,g^b=gij,g^c=gi,g^d=gij,g^e=gj,g^f=ghi

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238707
maximal normal subgroup of	#72238707

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);