#422876

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1921 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)(9,10,16,20)(11,15,19,14)(12,17,26,34)(13,27,31,22)(18,29,28,30)(21,24,33,25)(23,36,39,35)(32,38,40,37);(1,2,3,4)(9,11,14,10,16,19,15,20)(12,17,24,21,26,34,25,33)(13,28,29,27,31,18,30,22)(23,36,37,40,39,35,38,32);(5,6,7,8)(9,12)(10,17)(11,21)(13,23)(14,25)(15,24)(16,26)(18,32)(19,33)(20,34)(22,35)(27,36)(28,40)(29,38)(30,37)(31,39);(2,4)(9,13,16,31)(10,18,20,28)(11,22,19,27)(12,23,26,39)(14,30,15,29)(17,32,34,40)(21,35,33,36)(24,38,25,37)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,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=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

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, 330072);