#9947433

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854629);