#498157

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

788 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)(9,10,13,16)(11,17,21,33)(12,15,27,30)(14,28,26,19)(18,35,32,24)(20,38,25,31)(22,36,39,40)(23,37,29,34);(5,7,8,6)(10,15)(11,20)(12,24)(14,26)(16,30)(17,33)(18,19)(21,25)(22,29)(23,39)(27,35)(28,32)(31,40)(36,38);(2,4)(10,16)(11,21)(15,30)(18,32)(19,28)(20,25)(22,39)(23,29);(9,11,13,21)(10,17,16,33)(12,25,27,20)(14,29,26,23)(15,31,30,38)(18,36,32,40)(19,37,28,34)(22,35,39,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405353);