#6809846

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,30)(23,27,32,41)(24,29,36,46)(25,28,38,31)(33,40,35,44)(34,42,47,39)(37,43,50,45)(48,52,49,51);(2,3)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,35)(25,34)(26,32)(28,42)(29,44)(30,41)(31,39)(33,36)(37,49)(38,47)(40,46)(43,51)(45,52)(48,50);(1,2,4,3)(22,28)(23,32)(25,38)(27,39)(29,43)(30,31)(33,35)(37,50)(40,51)(41,42)(44,52)(45,46);(6,11)(10,13)(12,19)(18,20)(21,24,26,36)(22,29,30,46)(23,33,32,35)(25,37,38,50)(27,40,41,44)(28,43,31,45)(34,48,47,49)(39,51,42,52);(1,3,4,2)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,30)(23,32)(24,36)(28,31)(34,47)(37,50)(40,44)(51,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717042);