#6868307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775503);