#455386

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1118 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=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, 362582);