#494909

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

884 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)(5,6,8,7)(9,10,16,22)(11,23,21,14)(12,27,31,24)(13,19,17,26)(15,25,33,20)(18,34,37,30)(28,36,32,39)(29,40,38,35);(5,7,8,6)(9,11,15,26,17,25,14,10,16,21,33,19,13,20,23,22)(12,24,30,40,32,39,29,18,31,27,34,35,28,36,38,37);(2,4)(9,12,14,29,13,28,15,30,16,31,23,38,17,32,33,34)(10,18,20,36,26,40,21,27,22,37,25,39,19,35,11,24);(9,13,16,17)(10,19,22,26)(11,25,21,20)(12,28,31,32)(14,15,23,33)(18,35,37,40)(24,39,27,36)(29,30,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402105);