#423949

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331145);