#409661

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg

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, 316857);