#6912703

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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

1659 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)(19,24,30,39)(20,31,34,45)(21,26,22,36)(25,32,41,33)(28,42,44,47)(29,38,35,40)(37,46,48,43);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,40)(27,39)(31,42)(32,46)(33,43)(34,44)(36,38)(41,48)(45,47);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,32,23,34,22,33)(18,25,36,45,27,41,26,31)(19,28,35,46,30,44,29,43)(24,37,38,47,39,48,40,42);(2,6)(3,8)(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,34,33)(24,38,39,40)(25,31,41,45)(28,43,44,46)(37,47,48,42);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(19,30)(24,39)(28,44)(29,35)(37,48)(38,40)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=i,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=efhi,d^c=dfh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819899);