#6909666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6816862);