#498297

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	#392	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

689 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,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, 405493);