#467601

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374797);