#8181163

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

4511 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8088359);