#6465976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(4,5)(7,8)(9,10)(11,22)(12,26)(13,18)(14,19)(15,21)(16,24)(17,30)(20,33)(23,31)(25,32)(27,34)(28,39)(29,37)(35,38)(36,40);(2,5)(6,8)(9,11)(10,16)(12,27)(13,21)(14,24)(15,25)(17,28)(18,33)(19,22)(20,32)(23,30)(26,38)(29,35)(31,36)(34,37)(39,40);(9,12,14,29)(10,17)(11,23)(13,31,32,28)(15,30,33,40)(16,34,22,38)(18,27)(19,36)(20,26)(21,37)(24,39)(25,35);(9,13,15,16,14,32,33,22)(10,18,21,24,19,25,20,11)(12,28,30,38,29,31,40,34)(17,35,37,23,36,27,26,39);(1,3,4,7)(2,6,5,8)(12,29)(17,36)(23,39)(26,37)(27,35)(28,31)(30,40)(34,38)

Polycyclic presentation:

2^9|c^2=gh,d^2=h,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=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, 6373172);