Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454369 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1893 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)(5,6,11,15)(7,16,14,10)(8,12,21,28)(9,22,25,13)(17,20,31,26)(18,24,30,32)(19,27,34,33)(23,35,36,29);(5,7,10,6,11,14,16,15)(8,17,20,28,21,31,26,12)(9,18,24,13,25,30,32,22)(19,29,23,33,34,35,36,27);(2,4)(5,8)(6,12)(7,17)(9,23)(10,26)(11,21)(13,29)(14,31)(15,28)(16,20)(18,33)(19,24)(22,35)(25,36)(27,30)(32,34);(5,9,10,24,11,25,16,32)(6,13,14,30,15,22,7,18)(8,19,20,23,21,34,26,36)(12,27,17,29,28,33,31,35)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285552, #72285620, … |
| maximal normal subgroup of | #72285454, #72285552, #72285620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361566);