#458895

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,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, 366091);