#494092

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	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 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)(3,6)(4,5)(7,8)(9,10,15,21,16,23,29,22)(11,18,26,37,24,36,28,33)(12,17,25,34,27,32,40,35)(13,20,14,19,30,38,31,39);(2,5)(6,8)(9,11,14,25,15,26,30,27,16,24,31,40,29,28,13,12)(10,17,20,33,22,35,39,36,23,32,38,37,21,34,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,28,29,40,31,24,16,27,30,26,15,25,14,11)(10,18,19,34,21,37,38,32,23,36,39,35,22,33,20,17);(11,24)(12,27)(17,32)(18,36)(25,40)(26,28)(33,37)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=hi,f^2=h,g^2=i,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, 401288);