#6801539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 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)(17,18,23,37,22,27,21,26)(19,24,29,39,30,40,31,41)(20,32,34,25,36,43,35,44)(28,45,47,38,33,48,46,42);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19)(18,24)(20,33)(21,31)(22,30)(23,29)(25,42)(26,41)(27,40)(28,36)(32,48)(34,46)(35,47)(37,39)(38,44)(43,45);(10,13)(14,15)(17,20)(18,25)(19,28)(21,34)(22,36)(23,35)(24,38)(26,43)(27,44)(29,46)(30,33)(31,47)(32,37)(39,45)(40,42)(41,48);(2,6)(5,7)(17,21,22,23)(18,26,27,37)(19,29,30,31)(20,34,36,35)(24,39,40,41)(25,43,44,32)(28,46,33,47)(38,45,42,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,30)(24,40)(28,33)(29,31)(38,42)(39,41)(45,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708735);