#7183835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7091031);