#6696027

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(20,31,34,24)(22,26,35,42)(28,43,36,38)(29,39,33,41)(32,44,37,40)(45,47,46,48);(1,3,4,8)(2,5,6,7)(19,28)(20,32)(22,35)(23,38)(24,40)(26,42)(27,43)(30,36)(31,44)(34,37)(45,46)(47,48);(10,13)(14,16)(17,19,21,30)(18,23,25,27)(20,33,34,29)(22,36,35,28)(24,41,31,39)(26,43,42,38)(32,46,37,45)(40,48,44,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(20,34)(24,31)(29,33)(32,37)(39,41)(40,44)(45,46)(47,48);(3,8)(5,7)(17,20,21,34)(18,24,25,31)(19,29,30,33)(22,37,35,32)(23,39,27,41)(26,44,42,40)(28,45,36,46)(38,47,43,48)

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,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6603223);