#8445964

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

4480 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=i,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, 8353160);