#6432363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
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×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339559);