#420927

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1489 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:

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

Polycyclic presentation:

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

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, 328123);