#6501093

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1720 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)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,26,35,43)(25,31,41,28)(29,39,36,48)(32,46,34,47)(40,45,42,44);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,30,41,38)(26,39,43,48)(32,45,34,44)(40,46,42,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,34,35,32)(24,38,37,30)(26,42,43,40)(29,45,36,44)(39,46,48,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,35)(18,26,27,43)(19,29,22,36)(20,32,33,34)(24,39,37,48)(25,40,41,42)(28,44,31,45)(30,46,38,47);(10,14)(13,15)(21,35)(26,43)(29,36)(32,34)(39,48)(40,42)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6408289);