#6435076

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432953
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

3431 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 6342272);