#6741504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,30,39)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,29,38)(31,40,46,48)(33,42,34,43);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,23,26,39)(20,31,32,46)(21,28,35,29)(24,40,41,48)(25,37,44,38)(27,34,45,33)(36,43,47,42);(2,6)(5,7)(10,14)(15,16)(17,20,22,32)(18,24,26,41)(19,27,30,45)(21,34,35,33)(23,36,39,47)(25,43,44,42)(28,31,29,46)(37,40,38,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(20,32)(24,41)(27,45)(31,46)(33,34)(36,47)(40,48)(42,43);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,28)(21,35)(23,37)(25,44)(27,31)(29,30)(33,34)(36,40)(38,39)(42,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6648700);