#457805

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

931 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,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, 365001);