#6500227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62886	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407423);