#6685865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 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)(23,27,32,40)(24,34,36,49)(25,38,35,29)(28,42,43,37)(31,44,47,52)(33,48,45,41)(39,50,51,46);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,31)(25,33)(26,32)(28,39)(29,41)(30,40)(34,44)(35,45)(36,47)(37,46)(38,48)(42,50)(43,51)(49,52);(2,4)(21,24,35,42,26,36,25,37)(22,28,38,49,30,43,29,34)(23,31,45,50,32,47,33,46)(27,39,48,52,40,51,41,44);(6,11)(10,13)(12,19)(18,20)(23,32)(24,36)(27,40)(28,43)(33,45)(34,49)(37,42)(41,48);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593061);