#6868067

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

1493 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)(17,18,21,27,22,25,37,26)(19,28,31,40,30,23,38,41)(20,32,36,43,35,44,34,24)(29,45,47,39,33,48,46,42);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,25,28)(20,33,35,29)(21,31,37,38)(24,42,43,39)(26,40,27,41)(32,45,44,48)(34,47,36,46);(10,14)(15,16)(17,20)(18,24)(19,29)(21,34)(22,35)(23,39)(25,43)(26,32)(27,44)(28,42)(30,33)(31,46)(36,37)(38,47)(40,48)(41,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(19,30)(24,43)(26,27)(29,33)(31,38)(32,44)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,37)(18,26,25,27)(19,31,30,38)(20,34,35,36)(23,40,28,41)(24,32,43,44)(29,46,33,47)(39,48,42,45)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6775263);