#460079

order	512
nilpotency class	3
isoclinism	isoclinic class of #454369
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1167 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,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, 367275);