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 | |
1189 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1189 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:
(1,2,6,10)(3,12,8,15)(4,13,14,7)(5,11,16,9)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,31,32,29);(1,3,6,8)(2,4,10,14)(5,13,16,7)(9,15,11,12)(22,28)(29,31);(1,4)(2,7)(3,11)(5,12)(6,14)(8,9)(10,13)(15,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,32)(30,31);(1,4)(2,8)(3,10)(5,15)(6,14)(7,9)(11,13)(12,16)(17,20)(18,23)(19,25)(21,27)(22,29)(24,30)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=hi,b^2=h,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 | #72267806 |
| maximal normal subgroup of | #72267806 |
| maximal quotient of | #72267810 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151704);