Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417566 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,9)(7,13,20,18)(8,21,24,31)(10,16,11,15)(14,26,23,25)(17,30,34,36)(19,32,27,29)(22,28,35,33);(1,2)(3,4)(5,7)(6,13)(8,22)(9,18)(10,19)(11,27)(12,20)(14,30)(15,29)(16,32)(17,26)(21,33)(23,36)(24,35)(25,34)(28,31);(5,8)(6,14)(7,17)(9,23)(10,26)(11,25)(12,24)(13,28)(15,31)(16,21)(18,33)(19,35)(20,34)(22,27)(29,36)(30,32);(2,4)(5,9,12,6)(7,18,20,13)(8,23,24,14)(10,15,11,16)(17,33,34,28)(19,29,27,32)(21,26,31,25)(22,30,35,36)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72284191, #72284254, #72318083, … |
| maximal normal subgroup of | #72284191, #72284254, #72318083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324765);