#419220

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326416);