#475237

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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)(11,15)(12,13)(14,16)(25,26)(27,37)(28,32)(29,35)(30,33)(31,36)(34,39)(38,40);(2,5)(6,8)(10,13)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27,29,28,31,39,30,38)(26,32,33,37,36,40,35,34);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(18,21)(22,24)(26,33)(27,38)(28,39)(29,30)(34,37)(35,36);(3,7)(6,8)(11,16)(14,15)(17,19,20,23)(18,22,21,24)(25,28,30,27,31,38,29,39)(26,34,35,40,36,37,33,32)

Polycyclic presentation:

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

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, 382433);