#6822611

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,22,27)(19,24,30,39)(20,31,34,25)(21,26,37,44)(23,28,33,41)(29,45,47,38)(32,42,35,40)(36,48,46,43);(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,24,27,39)(20,32,34,35)(21,23,37,33)(25,40,31,42)(26,28,44,41)(29,46,47,36)(38,48,45,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,33,46,22,34,23,36)(18,25,41,48,27,31,28,43)(19,29,37,35,30,47,21,32)(24,38,44,42,39,45,26,40);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(20,34)(25,31)(29,47)(32,35)(36,46)(38,45)(40,42)(43,48);(10,14)(15,16)(17,21,23,30,22,37,33,19)(18,26,28,39,27,44,41,24)(20,35,36,29,34,32,46,47)(25,42,43,38,31,40,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=gi,e^2=h,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, 6729807);