#6508756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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×#3191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415952);