#232346

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 139542);