#9980311

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

4465 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,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, 9887507);