#6936860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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

1562 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)(17,18,23,37,22,28,21,27)(19,29,31,41,32,24,33,40)(20,26,34,25,35,42,36,43)(30,45,48,44,47,39,46,38);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,39)(27,41)(28,29)(34,48)(35,47)(36,46)(37,40)(42,45)(43,44);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,35)(18,25,28,43)(19,30,32,47)(21,34,23,36)(24,38,29,44)(26,27,42,37)(31,46,33,48)(39,41,45,40);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,39)(25,37)(27,43)(28,42)(29,45)(31,48)(32,47)(33,46)(38,40)(41,44);(10,13)(14,16)(17,21,22,23)(18,27,28,37)(19,31,32,33)(20,34,35,36)(24,40,29,41)(25,42,43,26)(30,46,47,48)(38,39,44,45)

Polycyclic presentation:

2^9|a^2=gi,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844056);