#453781

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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

844 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:

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

Polycyclic presentation:

2^9|a^2=efghi,b^2=efgi,c^2=efgi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=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, 360977);