#472866

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

803 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)(4,5)(7,8)(9,10,13,18)(11,20,24,15)(12,16,27,35)(14,28,21,19)(17,29,34,26)(22,33,23,30)(25,32,38,37)(31,40,36,39);(9,11,21,38,13,24,14,25)(10,15,28,37,18,20,19,32)(12,23,26,40,27,22,29,39)(16,33,34,36,35,30,17,31);(2,5)(6,8)(9,12)(10,16)(11,22,24,23)(13,27)(14,26)(15,30,20,33)(17,28)(18,35)(19,34)(21,29)(25,39,38,40)(31,37,36,32);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,23)(13,27)(14,29)(15,31)(16,28)(18,34)(19,35)(20,36)(21,26)(22,24)(25,39)(30,32)(33,37)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,g^d=gi,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, 380062);