#494904

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

856 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)(11,19)(12,18)(13,20)(14,22)(15,21)(16,32)(17,23)(24,34)(25,36)(26,40)(27,37)(28,33)(29,35)(30,38)(31,39);(5,7,8,6)(9,11,15,25,16,26,31,29,17,27,30,28,13,24,14,12)(10,18,22,34,20,33,38,37,23,35,39,40,32,36,21,19);(2,4)(9,12,14,24,13,28,30,27,17,29,31,26,16,25,15,11)(10,19,21,36,32,40,39,35,23,37,38,33,20,34,22,18);(9,13,17,16)(10,20,23,32)(11,24,27,26)(12,28,29,25)(14,30,31,15)(18,33,35,36)(19,34,37,40)(21,22,38,39)

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402100);