#9683701

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

4369 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)(11,16)(12,19)(13,21)(14,15)(17,23)(18,20)(22,24)(25,26,31,28)(27,32,34,29)(30,36,39,33)(35,37,40,38);(1,2)(3,8)(4,6)(5,7)(9,11,14,19)(10,12,15,16)(13,18,23,22)(17,20,21,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,38)(36,37)(39,40);(1,3,5,6)(2,4,7,8)(10,15)(12,16)(17,21)(20,24)(26,28)(27,34)(33,36)(35,40);(9,12,14,16)(10,11,15,19)(13,20,23,24)(17,18,21,22)(25,28,31,26)(27,29,34,32)(30,36,39,33)(35,37,40,38);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,29)(26,27)(28,34)(30,38)(31,32)(33,40)(35,36)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,23)(15,21)(16,24)(19,22)(25,30,31,39)(26,33,28,36)(27,35,34,40)(29,37,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^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, 9590897);