Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454369 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,22)(10,18)(11,14)(13,25)(16,20)(17,24)(19,27)(21,29)(23,32)(26,28)(30,35)(31,33)(34,36);(1,3,4,2)(5,7,15,6,11,18,10,14)(8,16,26,29,21,28,20,12)(9,17,31,13,25,33,24,22)(19,32,35,36,34,30,23,27);(2,3)(5,8)(6,12)(7,16)(9,23)(10,26)(11,21)(13,30)(14,29)(15,20)(17,27)(18,28)(19,24)(22,32)(25,35)(31,34)(33,36);(5,9,10,24,11,25,15,31)(6,13,7,17,14,22,18,33)(8,19,20,23,21,34,26,35)(12,27,28,30,29,36,16,32)Polycyclic presentation:
2^9|a^2=f,b^2=efi,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, #72285577, … |
| maximal normal subgroup of | #72285454, #72285552, #72285577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364428);