#6830291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,34)(22,27)(23,28)(26,35)(29,37)(30,38)(31,39)(32,40)(33,42)(36,43)(41,46)(44,47)(45,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,38)(20,31,32,46)(21,22,35,36)(25,39,40,41)(26,43,34,27)(29,44,45,33)(37,47,48,42);(2,5)(6,8)(10,14)(11,16)(17,20,22,33,23,32,36,44)(18,25,27,42,28,40,43,47)(19,29,21,31,30,45,35,46)(24,37,34,39,38,48,26,41);(10,14)(13,15)(20,32)(25,40)(29,45)(31,46)(33,44)(37,48)(39,41)(42,47);(1,3,4,7)(2,6,5,8)(17,21,22,30,23,35,36,19)(18,26,27,24,28,34,43,38)(20,31,33,45,32,46,44,29)(25,41,42,37,40,39,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6737487);