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×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2476 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)(5,6,11,16)(7,12,19,28)(8,20,22,31)(9,15,10,14)(13,24,30,23)(17,29,33,36)(18,32,25,27)(21,26,34,35);(1,2)(3,4)(5,7)(6,12)(8,21)(9,18)(10,25)(11,19)(13,29)(14,27)(15,32)(16,28)(17,24)(20,35)(22,34)(23,33)(26,31)(30,36);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,26)(14,31)(15,20)(16,30)(18,34)(19,33)(21,25)(27,36)(28,35)(29,32);(2,4)(9,10)(14,15)(18,25)(20,31)(21,34)(23,24)(27,32)(29,36)Polycyclic presentation:
2^9|a^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 | #72285179, #72285206, #72285237, … |
| maximal normal subgroup of | #72285179, #72285206, #72285237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324764);