#6807756

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1774 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,31)(23,27,32,41)(24,29,36,45)(25,28,38,30)(33,40,35,44)(34,42,47,39)(37,43,50,46)(48,52,49,51);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,26,32)(22,27,31,41)(24,35,36,33)(25,34,38,47)(28,42,30,39)(29,44,45,40)(37,49,50,48)(43,51,46,52);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(23,32)(25,38)(27,39)(29,43)(30,31)(33,35)(37,50)(40,51)(41,42)(44,52)(45,46);(6,12)(10,13)(11,18)(17,19)(21,24)(22,29)(23,33)(25,37)(26,36)(27,40)(28,43)(30,46)(31,45)(32,35)(34,48)(38,50)(39,51)(41,44)(42,52)(47,49);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(23,32)(24,36)(27,41)(29,45)(34,47)(37,50)(39,42)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,e^2=hi,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, 6714952);