#6797623

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6704819);