Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
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 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,16,11,15)(7,17,20,12,19,28,21,29)(8,22,25,31,26,30,24,13)(18,33,34,36,23,27,35,32);(1,2)(3,4)(5,7)(6,12,16,29)(8,23)(9,20)(10,19)(11,21)(13,27,31,33)(14,28,15,17)(18,26)(22,32,30,36)(24,34)(25,35);(2,4)(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,30)(15,22)(16,31)(17,32)(19,23)(20,35)(21,34)(28,36)(29,33);(6,14)(7,19)(8,24)(9,11)(12,28)(15,16)(17,29)(18,34)(22,30)(23,35)(25,26)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285454, #72285534, #72285613 |
| maximal normal subgroup of | #72285454, #72285534, #72285613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384221);