#6459459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366655);