#6694458

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=gi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,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, 6601654);