Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403943 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2752 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 2752 conjugacy classes of subgroups — too many to draw. Only its 240 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,8)(5,16)(6,10)(9,17)(11,18)(12,19)(13,20)(14,22)(15,23)(21,25)(24,31)(26,28)(27,29)(30,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,21)(9,19)(10,20)(14,24)(15,25)(16,28)(17,26)(18,23)(22,31)(27,32)(29,30);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,22)(10,23)(12,24)(13,25)(16,29)(17,27)(19,30)(20,21)(26,32)(28,31);(1,5)(2,9,10,16)(3,12)(4,14)(6,17)(7,19,20,28)(8,22,23,29)(11,24)(13,26)(15,27)(18,30,21,31)(25,32)Polycyclic presentation:
2^9|d^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234953, #72237842, #72237852, … |
| maximal normal subgroup of | #72234953, #72237842, #72237852, … |
| maximal quotient of | #72234956, #72234958, #72265870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311139);