#445128

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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)(3,6)(4,5)(7,8)(9,10,14,21,16,20,15,30)(11,22,12,18,25,17,24,33)(13,27,28,19,29,37,31,36)(23,38,26,35,39,34,40,32);(9,11,16,25)(10,17,20,22)(12,14,24,15)(13,23,29,39)(18,30,33,21)(19,35,36,32)(26,28,40,31)(27,34,37,38);(2,5)(6,8)(9,12,14,25,16,24,15,11)(10,18,21,17,20,33,30,22)(13,26,28,39,29,40,31,23)(19,34,37,32,36,38,27,35);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,23)(12,26)(14,28)(15,31)(16,29)(17,32)(18,34)(20,36)(21,37)(22,35)(24,40)(25,39)(27,30)(33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 352324);