#6564206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1485 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:

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

Polycyclic presentation:

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

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, 6471402);