Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2471 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)(7,12)(8,13)(9,15)(10,14)(11,16)(17,25)(18,26)(19,28)(20,35)(21,31)(22,30)(23,27)(24,32)(29,34)(33,36);(5,7)(6,12)(8,20)(9,23)(10,18)(11,19)(13,29)(14,26)(15,27)(16,28)(17,22)(21,34)(24,33)(25,32)(30,36)(31,35);(2,4)(5,8)(6,13)(7,17)(9,24)(10,22)(11,21)(12,25)(14,30)(15,32)(16,31)(18,34)(19,33)(20,23)(26,29)(27,35)(28,36);(6,14)(9,10)(12,26)(13,30)(15,16)(18,23)(20,34)(22,24)(25,29)(27,28)(31,32)(35,36)Polycyclic presentation:
2^9|g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255864, #72256282, #72256291, … |
| maximal normal subgroup of | #72255864, #72256282, #72256291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324818);