#423207

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1540 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)(10,16)(11,21)(13,14)(18,30)(19,20)(22,32)(23,37)(24,25)(26,34)(27,28)(31,38)(35,39);(5,6)(7,8)(9,10,13,16,15,20,14,19)(11,22,21,17,25,32,24,33)(12,26,27,34,29,30,28,18)(23,38,37,40,39,31,35,36);(1,2,3,4)(6,8)(9,11,15,25)(10,17,20,33)(12,23,29,39)(13,24,14,21)(16,22,19,32)(18,31,34,38)(26,40,30,36)(27,35,28,37);(9,12,15,29)(10,18,20,34)(11,23,25,39)(13,28,14,27)(16,30,19,26)(17,31,33,38)(21,35,24,37)(22,36,32,40)

Polycyclic presentation:

2^9|b^2=e,c^2=ef,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, 330403);