#434413

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=efi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 341609);