#8726379

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

4570 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)(9,10,15,13)(11,16,18,12)(14,17,23,21)(19,22,24,20)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,2,6,5)(3,8,4,7)(9,11,15,18)(10,16,13,12)(14,19,23,24)(17,22,21,20)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,3,6,4)(2,7,5,8)(11,18)(12,16)(19,24)(20,22)(28,37)(32,36)(35,38)(39,40);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,18)(14,22)(15,16)(17,24)(19,21)(20,23)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,5,6,2)(3,8,4,7)(9,13,15,10)(11,12,18,16)(14,17,23,21)(19,22,24,20)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(9,14,15,23)(10,17,13,21)(11,19,18,24)(12,20,16,22)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=h,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, 8633575);