#6915802

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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)(4,5)(7,8)(17,18,23,27)(19,28,32,43)(20,25,33,39)(21,26,22,36)(24,30,38,31)(29,42,44,48)(34,40,35,41)(37,46,47,45);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(20,33)(25,39)(29,44)(34,35)(37,47)(40,41)(42,48)(45,46);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,34,45,33,44,35,46)(25,37,41,42,39,47,40,48);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,42)(30,46)(31,45)(32,44)(36,40)(38,47)(43,48);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,43)(25,40,39,41)(29,45,44,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=fi,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^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, 6822998);