#72331754

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|d^2=h,b^a=bj,c^b=cg,d^b=dg,d^c=di,e^d=eg,f^a=fi,f^c=fi,f^d=fgj

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331772, #72332173, #72332391, …
maximal normal subgroup of	#72331772, #72332173, #72332391, …
maximal quotient of	#72333327

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