#72238245

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 15418 conjugacy classes of subgroups; only the 31753 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:

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

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=fi,f^b=fhj,f^c=fh,f^d=fhj,f^e=fj,g^b=gh,g^c=gh,g^d=gh,g^e=gi

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