#472864

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 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)(11,15)(12,16)(13,18)(14,17)(19,27)(20,28)(21,30)(22,29)(23,31)(24,32)(25,34)(26,35)(33,38)(36,39)(37,40);(9,11)(10,15,18,31)(12,24)(13,23)(14,22)(16,32,34,39)(17,29,19,30)(20,26)(21,27)(25,36)(28,33,40,35)(37,38);(2,5)(6,8)(9,12)(10,16)(11,20,23,37)(13,25)(14,26)(15,28,31,40)(17,35)(18,34)(19,33)(21,24,22,36)(27,38)(29,39,30,32);(1,3,4,7)(2,6,5,8)(10,17)(12,25)(14,27)(15,29)(16,33)(18,19)(21,22)(24,36)(28,39)(30,31)(32,40)(34,35)

Polycyclic presentation:

2^9|b^2=egh,c^2=ei,d^2=f,g^2=i,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^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, 380060);