#451185

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,21,23,24)(25,26,30,38,28,31,29,33)(27,34,37,32,35,40,36,39);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,23)(22,24);(1,4,5,2)(3,7,8,6)(10,14)(13,16)(19,23)(21,24)(26,31)(32,39)(33,38)(34,40);(3,8)(6,7)(25,27,28,35)(26,32,31,39)(29,37,30,36)(33,40,38,34)

Polycyclic presentation:

2^9|a^2=efgh,b^2=fi,c^2=e,d^2=i,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, 358381);