#6561865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fgi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6469061);