#6452426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

1643 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)(9,10)(11,16)(12,14)(13,15)(17,18,23,28)(19,24,32,40)(20,25,35,43)(21,26,22,27)(29,37,46,48)(30,38,31,39)(33,41,45,47)(34,42,36,44);(2,5)(6,8)(17,19,23,32)(18,24,28,40)(20,33,35,45)(21,31,22,30)(25,41,43,47)(26,39,27,38)(29,36,46,34)(37,44,48,42);(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,42)(27,44)(28,43)(30,33)(31,45)(32,46)(38,41)(39,47)(40,48);(9,12,13,11)(10,14,15,16)(19,30)(21,22)(24,38)(26,27)(29,33)(31,32)(34,36)(37,41)(39,40)(42,44)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,28,27)(19,31,32,30)(20,34,35,36)(24,39,40,38)(25,42,43,44)(29,45,46,33)(37,47,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359622);