#8726381

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

4559 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633577);