#8445922

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353118);