#6692774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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)(9,10,13,15)(11,14,12,16);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,33)(22,36)(23,27)(24,29)(26,32)(30,39)(31,38)(34,42)(35,41)(37,40)(43,45)(44,47)(46,48);(9,12,13,11)(10,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,44)(33,43)(34,45)(35,46)(41,47)(42,48);(2,6)(5,7)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,34,35,32)(24,38,40,28)(26,33,42,41)(30,45,46,44)(39,43,48,47);(3,8)(5,7)(17,21,23,35)(18,26,27,42)(19,30,31,46)(20,32,22,34)(24,39,40,48)(25,41,36,33)(28,43,38,47)(29,44,37,45)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6599970);