#474728

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

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

916 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)(3,7,8,6)(9,10,13,16)(11,14,12,15)(25,26,31,36)(27,37,39,33)(28,40,38,32)(29,35,30,34);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,23,24,21)(25,27,30,28,31,39,29,38)(26,32,35,33,36,40,34,37);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,16,14)(17,19,20,24)(18,21,22,23)(27,28)(29,30)(32,33)(34,35)(37,40)(38,39);(3,8)(6,7)(9,12,13,11)(10,15,16,14)(25,28,29,27,31,38,30,39)(26,33,34,32,36,37,35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=efghi,c^2=efgi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 381924);