#464798

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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:

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

Polycyclic presentation:

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

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, 371994);