#9910075

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

4470 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,32)(28,36)(29,30)(31,35)(33,39)(34,40)(37,38);(1,3,5,6)(2,4,7,8)(11,17)(14,21)(18,23)(22,24)(25,27)(26,32)(28,29)(30,36)(31,37)(33,40)(34,39)(35,38);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,31,39)(26,33,35,36)(27,34,37,29)(30,38,40,32);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,33)(27,29)(30,38)(31,39)(32,40)(34,37)(35,36);(1,4,5,8)(2,6,7,3)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29)(26,30)(27,28)(31,34)(32,36)(33,38)(35,40)(37,39);(1,2)(3,4)(5,7)(6,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,34)(27,36)(28,38)(29,35)(31,40)(32,39)(33,37)

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817271);