#478287

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

972 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,26,30,17)(13,19)(14,18)(15,20)(16,23)(22,35,38,37)(24,34)(25,33)(27,32,39,40)(28,31,36,29);(1,3)(2,5)(4,7)(6,8)(9,11,15,25)(10,16,20,34)(12,27,30,39)(13,24,14,23)(17,35,26,37)(18,33,19,21)(22,31,38,29)(28,32,36,40);(2,6)(5,8)(9,10)(11,21)(12,28)(13,19)(14,18)(15,20)(16,23)(17,29)(22,32)(24,34)(25,33)(26,31)(27,37)(30,36)(35,39)(38,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,31)(14,29)(15,30)(16,32)(18,36)(19,28)(20,26)(21,37)(23,39)(24,27)(25,38)(33,35)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385483);