#474275

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

992 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)(11,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,26,32,38)(27,35,28,34)(29,33,39,40)(30,37,31,36);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19,20,23)(18,21,22,24)(25,27,31,39,32,28,30,29)(26,33,37,34,38,40,36,35);(1,3)(2,6)(4,8)(5,7)(10,15)(14,16)(19,23)(21,24)(25,28,32,27)(26,34,38,35)(29,31,39,30)(33,36,40,37);(3,8)(6,7)(9,12,13,11)(10,16,15,14)(25,29,30,28,32,39,31,27)(26,35,36,40,38,34,37,33)

Polycyclic presentation:

2^9|a^2=eg,b^2=efghi,c^2=i,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, 381471);