#72330693

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#15257	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7022 conjugacy classes of subgroups; only the 1079 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|c^2=hi,d^2=fi,e^2=i,b^a=bj,c^a=cf,c^b=cfh,d^a=df,d^b=dhi,d^c=dfh,e^a=eg,e^b=ehi,e^d=eg

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331120, #72331156, #72331614
maximal normal subgroup of	#72331120, #72331156, #72331614

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