Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394275 |
| 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 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2613 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)(7,15)(8,12)(9,22)(10,14)(11,19)(13,25)(16,23)(17,33)(18,29)(20,30)(21,28)(24,27)(26,35)(31,34)(32,36);(1,2,3,4)(5,6)(7,16,18,11)(8,12)(9,13)(10,14)(15,19,29,23)(17,32,34,26)(20,27)(21,28)(22,25)(24,30)(31,36,33,35);(5,7)(6,11)(8,17)(9,23)(10,18)(12,26)(13,29)(14,16)(15,22)(19,25)(20,36)(21,34)(24,35)(27,31)(28,32)(30,33);(5,8,10,21)(6,12,14,28)(7,17,18,34)(9,24,25,20)(11,26,16,32)(13,30,22,27)(15,31,29,33)(19,36,23,35)Polycyclic presentation:
2^9|b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318090, #72318110, #72318152, … |
| maximal normal subgroup of | #72318090, #72318110, #72318152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301472);