#460617

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,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, 367813);