#430415

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1236 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,d^2=efi,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, 337611);