#6828145

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,29,35)(20,24,31,37)(21,25,32,38)(27,33,40,44)(28,34,41,45)(30,36,42,46)(39,43,47,48);(1,3,6,2)(4,8,5,7)(19,27)(20,30)(21,32)(23,33)(24,36)(25,38)(28,41)(29,40)(31,42)(34,45)(35,44)(37,46);(10,13)(14,16)(17,19,21,27,22,29,32,40)(18,23,25,33,26,35,38,44)(20,28,30,39,31,41,42,47)(24,34,36,43,37,45,46,48);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(20,31)(24,37)(28,41)(30,42)(34,45)(36,46)(39,47)(43,48);(1,5,6,4)(2,7,3,8)(17,20,32,42,22,31,21,30)(18,24,38,46,26,37,25,36)(19,28,40,47,29,41,27,39)(23,34,44,48,35,45,33,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=fg,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6735341);