#6833124

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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)(19,29)(20,32)(21,27)(22,26)(23,28)(24,31)(25,35)(30,38)(33,40)(34,41)(36,47)(37,39)(42,48)(43,45)(44,46);(1,2)(3,6)(4,5)(7,8)(17,19,22,31)(18,24,26,29)(20,33,35,48)(21,23,37,34)(25,40,32,42)(27,41,39,28)(30,45,46,36)(38,47,44,43);(10,13)(14,16)(17,20,34,45,22,35,23,36)(18,25,41,47,26,32,28,43)(19,30,37,48,31,46,21,33)(24,38,27,42,29,44,39,40);(1,3,4,7)(2,6,5,8)(18,26)(20,35)(24,29)(27,39)(28,41)(30,46)(33,48)(36,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,34,19,22,37,23,31)(18,27,41,29,26,39,28,24)(20,33,45,30,35,48,36,46)(25,42,47,44,32,40,43,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,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,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, 6740320);