#440969

order	512
nilpotency class	3
isoclinism	isoclinic class of #419461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348165);