#8464259

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

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=h,e^2=i,f^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, 8371455);