#6833231

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6740427);