#6432895

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4329 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26)(13,18)(14,20)(15,19)(16,24)(17,27)(22,30)(23,31)(25,34)(28,32)(29,33)(35,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,21)(19,31)(20,34)(26,39)(27,36)(28,38)(29,37)(32,35)(33,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,32)(19,33)(20,26)(21,35)(23,36)(24,37)(25,38)(31,40)(34,39);(2,6)(5,8)(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,21,31,34)(17,32,33,26)(22,36,38,37)(30,35,40,39);(10,19)(16,31)(17,33)(18,20)(21,34)(26,32)(30,40)(35,39)

Polycyclic presentation:

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

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, 6340091);