#6514066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421262);