#6936918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1567 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)(4,5)(7,8)(17,18,24,30,23,29,22,28)(19,31,33,45,34,25,39,42)(20,27,35,43,21,36,38,26)(32,41,48,40,37,44,47,46);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,34)(18,25,29,31)(20,32,21,37)(22,39,24,33)(26,40,43,46)(27,44,36,41)(28,45,30,42)(35,48,38,47);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20)(18,26)(19,32)(21,23)(22,35)(24,38)(25,40)(27,28)(29,43)(30,36)(31,46)(33,47)(34,37)(39,48)(41,42)(44,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,36)(19,32,34,37)(22,35,24,38)(25,41,31,44)(26,30,43,28)(33,48,39,47)(40,45,46,42);(10,13)(14,16)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,35,21,38)(25,42,31,45)(26,27,43,36)(32,47,37,48)(40,41,46,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844114);