#72334930

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|f^2=h,b^a=bj,c^b=ch,d^a=dh,d^c=dh,f^b=fi,f^c=fj,f^d=fg,f^e=fj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334931, #72339952
maximal normal subgroup of	#72334931, #72339952
maximal quotient of	#72334982, #72335069, #72339978

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