#420767

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327963);