#433769

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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

944 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)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,19)(18,21)(20,23)(22,24)(25,27,30,29)(26,31,34,33)(28,35,37,36)(32,38,40,39);(1,2,5,4)(3,6,8,7)(10,14)(13,16)(19,23)(21,24)(25,28,30,37)(26,32,34,40)(27,35,29,36)(31,38,33,39);(1,4,5,2)(3,6,8,7)(26,33)(27,29)(31,34)(32,39)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=efg,b^2=efi,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340965);