#9947275

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

4355 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=g,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, 9854471);