#6834513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,26)(19,29,31,24)(20,32,34,25)(21,27,38,40)(23,28,37,43)(30,39,47,45)(33,41,35,44)(36,48,46,42);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,26,29)(20,33,34,35)(21,23,38,37)(25,41,32,44)(27,43,40,28)(30,46,47,36)(39,48,45,42);(9,11,13,12)(10,14,15,16)(17,20,23,36,22,34,37,46)(18,25,28,42,26,32,43,48)(19,30,21,33,31,47,38,35)(24,39,40,41,29,45,27,44);(2,6)(5,7)(18,26)(20,34)(24,29)(27,40)(28,43)(30,47)(33,35)(36,46);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,21,37,19,22,38,23,31)(18,27,43,29,26,40,28,24)(20,35,46,47,34,33,36,30)(25,41,48,39,32,44,42,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,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,e^c=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, 6741709);