#6716938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,29,38)(20,24,31,40)(21,25,35,44)(27,36,45,47)(28,37,33,42)(30,39,46,48)(32,41,34,43);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,30)(21,33)(22,29)(24,39)(25,42)(26,38)(27,32)(28,35)(31,46)(34,45)(36,41)(37,44)(40,48)(43,47);(10,15)(14,16)(17,20,22,31)(18,24,26,40)(19,27,29,45)(21,34,35,32)(23,36,38,47)(25,43,44,41)(28,46,33,30)(37,48,42,39);(1,3,4,7)(2,6,5,8)(20,31)(24,40)(27,45)(30,46)(32,34)(36,47)(39,48)(41,43);(2,5)(6,8)(9,12,13,11)(10,16,15,14)(19,28)(21,35)(23,37)(25,44)(27,46)(29,33)(30,45)(32,34)(36,48)(38,42)(39,47)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6624134);