#6833715

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,d^2=fi,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, 6740911);