#460613

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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)(9,10,15,21)(11,18,12,24)(13,27)(14,32,16,20)(17,26,34,23)(19,29)(22,38)(25,33)(28,35)(30,37)(31,36)(39,40);(1,3,4,7)(2,5,6,8)(9,11,16,26,15,12,14,23)(10,17,32,24,21,34,20,18)(13,22,29,39,31,25,30,28)(19,33,27,40,37,38,36,35);(2,6)(5,8)(9,12,15,11)(10,18,21,24)(13,28,31,39)(14,23,16,26)(17,32,34,20)(19,33,37,38)(22,29,25,30)(27,35,36,40);(9,13)(10,19)(11,22)(12,25)(14,30)(15,31)(16,29)(17,33)(18,35)(20,36)(21,37)(23,28)(24,40)(26,39)(27,32)(34,38)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,e^2=i,g^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,g^c=gi,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, 367809);