#6818168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(9,10,12,14)(11,15,16,13)(19,27)(23,36)(28,41)(29,42)(30,37)(38,44)(43,48)(45,47);(2,5)(3,7)(17,18,21,25)(19,28,27,41)(20,31,33,46)(22,35,32,26)(23,37,36,30)(24,39,40,34)(29,44,42,38)(43,47,48,45);(9,11,12,16)(10,13,14,15)(17,19,22,30,21,27,32,37)(18,23,26,28,25,36,35,41)(20,29,34,45,33,42,39,47)(24,38,31,43,40,44,46,48);(2,5)(6,8)(19,27)(23,36)(28,41)(29,42)(30,37)(38,44)(43,48)(45,47);(1,3,4,7)(2,6,5,8)(17,20,32,39,21,33,22,34)(18,24,35,46,25,40,26,31)(19,29,37,47,27,42,30,45)(23,38,41,48,36,44,28,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725364);