Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2760 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)(6,12)(9,10)(13,16)(14,27)(15,24)(18,25)(20,30)(21,23)(26,29)(28,31)(32,36)(33,34);(5,6,9,12,11,15,10,24)(7,16,18,13,19,26,25,29)(8,20,21,30,22,27,23,14)(17,32,33,36,35,28,34,31);(5,7,11,19)(6,13,15,29)(8,17,22,35)(9,18,10,25)(12,26,24,16)(14,28,30,32)(20,31,27,36)(21,33,23,34);(2,4)(5,8)(6,14)(7,17)(9,23)(10,21)(11,22)(12,27)(13,28)(15,30)(16,31)(18,34)(19,35)(20,24)(25,33)(26,36)(29,32)Polycyclic presentation:
2^9|b^2=e,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318739, #72318775, #72318894, … |
| maximal normal subgroup of | #72318739, #72318775, #72318894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324995);