Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,16,10,17)(7,18,20,30,22,31,21,12)(8,23,25,33,26,13,27,14)(19,29,24,34,35,32,36,28);(1,2)(3,4)(5,7,11,22)(6,12,16,30)(8,24,26,36)(9,21,10,20)(13,32,23,29)(14,34,33,28)(15,31,17,18)(19,27,35,25);(2,4)(5,8)(6,13)(7,19)(9,27)(10,25)(11,26)(12,28)(14,17)(15,33)(16,23)(18,32)(20,36)(21,24)(22,35)(29,31)(30,34);(5,8)(6,14,16,33)(7,19)(9,27)(10,25)(11,26)(12,29,30,32)(13,17,23,15)(18,28,31,34)(20,36)(21,24)(22,35)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334637);