#9851154

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

4343 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9758350);