#460921

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

950 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)(9,10,13,16,15,20,14,19)(11,21,23,17,25,32,24,34)(12,26,28,36,30,18,29,31)(22,35,27,33,38,39,37,40);(1,3,4,7)(2,5,6,8)(10,16,20,19)(17,32,34,21)(18,31,26,36)(33,39,40,35);(2,6)(5,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,21)(18,35)(19,32)(20,34)(22,28)(26,39)(29,38)(30,37)(31,40)(33,36);(9,12,13,28,15,30,14,29)(10,18,19,36,20,26,16,31)(11,22,23,27,25,38,24,37)(17,33,21,35,34,40,32,39)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 368117);