#6695835

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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1567 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)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,24,32,39)(22,26,33,40)(28,41,34,36)(29,43,45,37)(31,38,35,42)(44,48,46,47);(1,3,4,8)(2,5,6,7)(19,28)(20,31)(22,33)(23,36)(24,38)(26,40)(27,41)(30,34)(32,35)(39,42)(44,46)(47,48);(10,13)(14,16)(17,19,21,30)(18,23,25,27)(20,29,32,45)(22,34,33,28)(24,37,39,43)(26,41,40,36)(31,44,35,46)(38,47,42,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(20,32)(24,39)(29,45)(31,35)(37,43)(38,42)(44,46)(47,48);(3,8)(5,7)(17,20,21,32)(18,24,25,39)(19,29,30,45)(22,35,33,31)(23,37,27,43)(26,42,40,38)(28,44,34,46)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=gi,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, 6603031);