#6912677

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	#23929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,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, 6819873);