#6500025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500015
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	C₂⁴	C₂⁵
prime radical	#62862	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3330 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)(11,15,21,29)(12,22)(13,17,26,32)(16,24)(19,27)(20,28,25,31)(23,36)(30,38)(33,39)(34,35)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,25)(14,21)(16,27)(17,31)(18,29)(20,26)(22,35)(23,37)(24,34)(28,32)(30,40)(33,38)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,30)(18,22)(20,33)(21,34)(25,37)(26,38)(28,39)(29,35)(31,40)(32,36);(3,8)(5,7)(9,13,14,26)(10,17,18,32)(11,20,21,25)(12,23,24,38)(15,28,29,31)(16,30,22,36)(19,33,34,37)(27,39,35,40);(2,6)(5,7)(13,26)(17,32)(20,25)(23,38)(28,31)(30,36)(33,37)(39,40)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6407221);