#6842000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,29,31,24)(20,25,34,42)(21,27,37,39)(23,28,33,41)(30,45,47,38)(32,44,35,40)(36,43,46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,32,34,35)(21,23,37,33)(25,40,42,44)(27,41,39,28)(30,46,47,36)(38,48,45,43);(2,6)(7,8)(10,14)(11,16)(17,20,33,46,22,34,23,36)(18,25,41,48,26,42,28,43)(19,30,37,35,31,47,21,32)(24,38,27,44,29,45,39,40);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,26)(20,34)(24,29)(27,39)(28,41)(30,47)(32,35)(36,46);(1,4,5,3)(2,8,6,7)(17,21,33,19,22,37,23,31)(18,27,41,29,26,39,28,24)(20,35,46,47,34,32,36,30)(25,40,48,38,42,44,43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

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, 6749196);