#6907821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#30513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(3,6)(4,8)(5,7)(17,18,22,38,23,29,24,28)(19,30,39,43,33,25,32,42)(20,34,37,26,21,27,35,44)(31,45,47,46,36,41,48,40);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,25,29,30)(20,31,21,36)(22,39,24,32)(26,40,44,46)(27,45,34,41)(28,43,38,42)(35,48,37,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,35)(24,37)(25,40)(27,28)(29,44)(30,46)(32,47)(33,36)(34,38)(39,48)(41,42)(43,45);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,21,23,20)(18,27,29,34)(19,31,33,36)(22,35,24,37)(25,41,30,45)(26,38,44,28)(32,48,39,47)(40,43,46,42);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,22,23,24)(18,28,29,38)(19,32,33,39)(20,35,21,37)(25,42,30,43)(26,27,44,34)(31,47,36,48)(40,41,46,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815017);