#6512741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499849
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3104 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6419937);