#9684005

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9591201);