#422894

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1885 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)(9,10,16,22)(11,18,25,33)(12,17,24,34)(13,27,30,38)(14,21,15,20)(19,31,37,29)(23,32,39,40)(26,36,28,35);(1,2,3,4)(5,6)(7,8)(9,11,15,24,16,25,14,12)(10,17,21,33,22,34,20,18)(13,28,31,23,30,26,29,39)(19,36,38,32,37,35,27,40);(6,8)(9,12,14,25,16,24,15,11)(10,18,20,34,22,33,21,17)(13,26,31,39,30,28,29,23)(19,35,38,40,37,36,27,32);(2,4)(9,13)(10,19)(11,23)(12,26)(14,31)(15,29)(16,30)(17,32)(18,35)(20,38)(21,27)(22,37)(24,28)(25,39)(33,36)(34,40)

Polycyclic presentation:

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

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, 330090);