#422828

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,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, 330024);