#9848545

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

4334 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=hi,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, 9755741);