#8695842

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

4629 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)(3,4)(5,7)(6,8)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(27,34)(30,32)(35,40)(38,39);(1,3,6,5)(2,7,8,4)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,37)(38,40);(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(2,8)(3,5)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,28,29,37)(26,31,33,36)(27,35,34,40)(30,38,32,39);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,27)(26,32)(28,35)(29,34)(30,33)(31,39)(36,38)(37,40);(1,5,6,3)(2,7,8,4)(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,26)(27,30)(28,36)(29,33)(31,37)(32,34)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603038);