#6462900

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370096);