#422885

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1977 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)(6,8)(9,10,15,20)(11,21,25,32)(12,17,29,36)(13,19,14,18)(16,24,31,23)(22,33,38,40)(26,30,37,39)(27,34,28,35);(1,2,3,4)(9,11,14,24,15,25,13,23)(10,16,19,32,20,31,18,21)(12,26,28,38,29,37,27,22)(17,33,34,39,36,40,35,30);(5,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,30)(18,34)(19,35)(20,36)(21,33)(23,26)(24,37)(25,38)(31,39)(32,40);(2,4)(10,18)(11,23)(13,14)(16,31)(17,34)(19,20)(22,26)(24,25)(27,28)(30,39)(35,36)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330081);