Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,18)(8,15,26,14)(9,27,25,35)(10,29)(11,16)(12,17)(13,21)(19,34,28,33)(20,31,24,30)(22,32)(23,36);(1,2)(3,4)(5,7,12,23)(6,13,17,32)(8,24,26,20)(9,28,25,19)(10,22,11,21)(14,33,15,34)(16,36,29,18)(27,30,35,31);(5,8,11,25,12,26,10,9)(6,14,16,27,17,15,29,35)(7,19,22,24,23,28,21,20)(13,30,18,33,32,31,36,34);(1,3,4,2)(5,9,10,26,12,25,11,8)(6,15,16,35,17,14,29,27)(7,20,21,28,23,24,22,19)(13,31,18,34,32,30,36,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306448, #72306459, #72306464 |
| maximal normal subgroup of | #72306448, #72306459, #72306464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293229);