#6674516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#29607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6581712);