#72238255

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 10770 conjugacy classes of subgroups; only the 31745 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=h,f^2=j,g^2=i,b^a=bhi,c^b=ch,d^a=dh,d^c=dh,f^a=fij,f^b=fi,f^c=fi,f^d=fi,f^e=fhi,g^a=gi,g^b=gi,g^d=gi,g^e=gh

Conjugacy classes & character table

178 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, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
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, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
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);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);