#495693

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=eh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402889);