#6567266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,32)(25,35)(29,43)(30,42)(31,39)(34,40)(36,47)(38,44)(41,46)(45,48);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,39,37,42)(20,29,35,46,36,44,34,45)(25,38,33,41,40,43,47,48);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,22,35,23,36,21,34)(18,25,26,33,27,40,37,47)(19,29,32,46,30,44,31,45)(24,38,28,41,39,43,42,48);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(19,30)(24,39)(28,42)(29,44)(31,32)(38,43)(41,48)(45,46);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,27,37)(19,31,30,32)(20,34,36,35)(24,28,39,42)(25,33,40,47)(29,45,44,46)(38,41,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fgi,e^2=fg,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^d=eh

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, 6474462);