#72331409

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

29837 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=gi,e^2=hij,b^a=bh,c^b=cf,d^a=dg,d^b=dhj,e^a=ehj,e^d=ef

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331504, #72331532, #72331630
maximal normal subgroup of	#72331504, #72331532, #72331630

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