#6936812

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

1569 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)(17,18,23,29,22,28,21,27)(19,30,32,43,33,24,37,41)(20,34,35,42,36,26,38,25)(31,40,48,39,47,45,46,44);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,31)(21,37)(22,33)(23,32)(25,39)(26,40)(27,43)(28,30)(29,41)(34,45)(35,48)(36,47)(38,46)(42,44);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,36)(18,25,28,42)(19,31,33,47)(21,35,23,38)(24,39,30,44)(26,27,34,29)(32,46,37,48)(40,43,45,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,31)(21,38)(22,36)(23,35)(24,40)(25,29)(27,42)(28,34)(30,45)(32,48)(33,47)(37,46)(39,41)(43,44);(10,13)(14,16)(17,21,22,23)(18,27,28,29)(19,32,33,37)(20,35,36,38)(24,41,30,43)(25,34,42,26)(31,46,47,48)(39,40,44,45)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6844008);