#6432001

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3414 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:

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

Polycyclic presentation:

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

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, 6339197);