#6833427

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,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, 6740623);