#494916

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 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)(9,10,16,23)(11,19,26,38)(12,18,28,35)(13,20,17,32)(14,22,31,40)(15,21,30,39)(24,37,27,34)(25,33,29,36);(5,7,8,6)(9,11,15,25,17,24,31,28,16,26,30,29,13,27,14,12)(10,18,22,34,20,36,39,38,23,35,40,37,32,33,21,19);(2,4)(9,12,14,27,13,29,30,26,16,28,31,24,17,25,15,11)(10,19,21,33,32,37,40,35,23,38,39,36,20,34,22,18);(9,13,16,17)(10,20,23,32)(11,24,26,27)(12,25,28,29)(14,30,31,15)(18,33,35,36)(19,37,38,34)(21,22,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,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,d^c=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, 402112);