#232348

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,9,14)(7,15,18,12)(8,11,10,13)(16,20,17,19)(21,22,27,30)(23,32,36,48)(24,29,39,45)(25,40,26,31)(28,35,43,34)(33,49,50,44)(37,42,51,52)(38,47,41,46);(6,11)(7,16)(8,10)(13,14)(17,18)(19,20)(21,23,27,36)(22,28,30,43)(24,37,39,51)(25,35,26,34)(29,44,45,49)(31,48,40,32)(33,41,50,38)(42,46,52,47);(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,20)(14,15)(21,24)(22,29)(23,33)(25,38)(26,41)(27,39)(28,42)(30,45)(31,46)(32,49)(34,37)(35,51)(36,50)(40,47)(43,52)(44,48);(5,8,9,10)(6,13,14,11)(7,16,18,17)(12,20,15,19)(22,30)(28,43)(29,45)(31,40)(32,48)(42,52)(44,49)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139544);