#498111

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=dei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405307);