Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419794 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,12,11,16,9,15)(7,17,19,29,20,26,25,13)(8,21,23,30,24,14,22,27)(18,32,33,28,34,31,36,35);(6,12)(8,22)(9,10)(13,17)(15,16)(18,33)(19,25)(23,24)(26,29)(27,30)(28,35)(34,36);(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,26)(14,28)(15,17)(16,29)(21,35)(22,33)(23,36)(24,34)(27,32)(30,31);(2,4)(5,8,9,22,11,24,10,23)(6,14,15,30,16,21,12,27)(7,18,19,33,20,34,25,36)(13,28,17,31,29,35,26,32)Polycyclic presentation:
2^9|a^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252171, #72256284, #72256289 |
| maximal normal subgroup of | #72252171, #72256284, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326990);