#6487663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470231
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

4175 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)(3,6)(4,7)(5,8)(9,10,14,18)(11,15,20,28)(12,16,24,32)(13,25,26,17)(19,30,22,27)(21,35,34,29)(23,37,38,31)(33,39,36,40);(3,4)(6,7)(9,11)(10,15)(12,22)(13,21)(14,20)(16,30)(17,29)(18,28)(19,24)(23,36)(25,35)(26,34)(27,32)(31,40)(33,38)(37,39);(1,3,5,4)(2,6,8,7)(9,12,14,24)(10,16,18,32)(11,19,20,22)(13,23,26,38)(15,27,28,30)(17,31,25,37)(21,33,34,36)(29,39,35,40);(1,4,5,3)(2,7,8,6)(11,20)(13,26)(15,28)(17,25)(19,22)(23,38)(27,30)(31,37);(2,8)(6,7)(9,13,14,26)(10,17,18,25)(11,21,20,34)(12,23,24,38)(15,29,28,35)(16,31,32,37)(19,33,22,36)(27,39,30,40)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,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, 6394859);