#6458984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,32)(21,26)(22,36)(23,27)(24,30)(25,34)(29,43)(31,39)(33,47)(35,41)(37,40)(38,48)(42,45)(44,46);(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,27,40)(20,29,34,45)(21,30,22,37)(25,38,32,46)(26,39,36,28)(33,44,35,48)(41,43,47,42);(2,5)(6,8)(11,15)(13,16)(17,20,23,34)(18,25,27,32)(19,29,31,45)(21,33,22,35)(24,38,40,46)(26,41,36,47)(28,42,39,43)(30,44,37,48);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,47)(29,44,45,48)(38,43,46,42);(1,3,4,7)(2,6,5,8)(20,34)(25,32)(29,45)(33,35)(38,46)(41,47)(42,43)(44,48)

Polycyclic presentation:

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

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, 6366180);