Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393933 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4216 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 4216 conjugacy classes of subgroups — too many to draw. Only its 296 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)(5,9)(6,10)(11,17)(12,26)(13,19)(14,28)(15,22)(16,23)(18,27)(20,25)(21,29)(24,30)(31,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,20)(9,18)(10,19)(14,24)(15,25)(16,27)(17,22)(21,31)(23,26)(28,30)(29,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,29)(18,30)(19,20)(23,28)(26,31)(27,32);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,27)(15,29)(17,30)(19,26)(20,31)(22,28)(25,32)Polycyclic presentation:
2^9|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=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72210507, #72234172, #72234953, … |
| maximal normal subgroup of | #72210507, #72234172, #72234953, … |
| maximal quotient of | #72234155, #72234959, #72240606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301129);