Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1087 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1087 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,12)(5,10)(6,9)(7,14)(8,16)(13,15)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3)(2,4)(5,7)(6,13)(8,15)(9,14)(10,11)(12,16)(17,18,22,26)(19,24,28,32)(20,25,21,23)(27,30,29,31);(1,4)(2,7)(3,8)(5,11)(6,14)(9,12)(10,13)(15,16)(18,23)(20,21)(24,30)(25,26)(27,29)(31,32);(1,4)(2,3)(5,15)(6,14)(7,8)(9,13)(10,12)(11,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|b^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267807 |
| maximal normal subgroup of | #72267807 |
| maximal quotient of | #72267810 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151709);