#417818

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2780 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)(3,4)(5,6)(7,8)(9,10,14,18)(11,15,21,30)(12,16,22,31)(13,24,27,17)(19,33,25,28)(20,29,36,39)(23,34,26,32)(35,40,37,38);(11,19,21,25)(12,22)(15,28,30,33)(16,31)(20,35,36,37)(23,26)(29,38,39,40)(32,34);(2,4)(9,11)(10,15)(12,20)(13,25)(14,21)(16,29)(17,33)(18,30)(19,27)(22,36)(23,35)(24,28)(26,37)(31,39)(32,38)(34,40);(6,8)(9,12,14,22)(10,16,18,31)(11,20,21,36)(13,26,27,23)(15,29,30,39)(17,34,24,32)(19,35,25,37)(28,38,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,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, 325014);