#453840

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

649 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)(6,8)(9,10)(11,20,24,34)(12,16,26,32)(13,17)(14,19)(15,23,29,22)(18,27)(21,36)(25,31,40,33)(28,38)(30,37)(35,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,21)(13,22)(14,24)(16,30)(17,34)(18,20)(19,29)(23,27)(25,37)(26,39)(28,32)(31,35)(33,36)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,25,14,26,27,40)(10,16,17,31,19,32,18,33)(11,21,23,38,24,39,22,37)(15,28,20,35,29,30,34,36);(2,5)(6,8)(10,17,19,18)(11,22,24,23)(15,29)(16,31,32,33)(20,34)(21,37,39,38)(28,30)(35,36)

Polycyclic presentation:

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

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, 361036);