#6830252

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	#42673	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,e^2=gh,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, 6737448);