Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2355 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)(7,19)(8,23)(9,28)(10,17)(11,16)(12,18)(13,20)(14,25)(15,24)(21,33)(22,32)(26,34)(27,31)(29,36)(30,35);(5,7)(6,13)(8,24)(9,25)(10,21)(11,20)(12,22)(14,15)(16,32)(17,19)(18,33)(23,35)(26,30)(27,29)(28,31)(34,36);(2,4)(5,8)(6,14)(7,9)(10,25)(11,26)(12,27)(13,28)(15,32)(16,23)(17,31)(18,34)(19,36)(20,24)(21,29)(22,30)(33,35);(5,9)(6,15)(7,8)(10,29)(11,24)(12,30)(13,31)(14,32)(16,35)(17,28)(18,36)(19,34)(20,26)(21,25)(22,27)(23,33)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322978 |
| maximal normal subgroup of | #72322978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324909);