#415520

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322716);