Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,14,10,15)(7,17,19,28,21,31,20,12)(8,22,26,13,25,30,24,29)(18,27,34,36,23,35,33,32);(1,2)(3,4)(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,27)(14,28)(15,31)(16,17)(18,25)(22,32)(24,34)(26,33)(29,35)(30,36);(2,4)(5,8)(6,13)(7,18,21,23)(9,24)(10,26)(11,25)(12,27,28,35)(14,29)(15,30)(16,22)(17,32,31,36)(19,33,20,34);(6,14)(7,19)(9,10)(12,17)(13,29)(18,33)(20,21)(23,34)(24,26)(27,32)(28,31)(35,36)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319962, #72319988, #72320018 |
| maximal normal subgroup of | #72319962, #72319988, #72320018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296228);