#6915483

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#30704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(17,18,22,26)(19,27,28,42)(20,24,32,40)(21,34,35,25)(23,36,37,30)(29,43,31,46)(33,48,44,41)(38,47,39,45);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(19,28)(23,37)(27,42)(29,31)(30,36)(38,39)(43,46)(45,47);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,28,35,36)(18,23,25,27,26,37,34,42)(20,31,33,47,32,29,44,45)(24,39,41,46,40,38,48,43);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,20,22,32)(18,24,26,40)(19,29,28,31)(21,33,35,44)(23,38,37,39)(25,41,34,48)(27,43,42,46)(30,45,36,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,32)(24,40)(29,31)(33,44)(38,39)(41,48)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822679);