#9874911

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

4599 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gi,e^2=gi,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, 9782107);