Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245282 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1857 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1857 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,8)(6,11)(7,15)(10,18)(13,26)(14,25)(16,23)(17,22)(19,31)(20,30)(21,29)(24,32)(27,28);(4,16)(7,19)(9,23)(12,25)(15,29)(17,32)(20,21)(22,24)(26,27)(30,31);(1,3)(2,7)(4,15)(5,14)(6,13)(8,20)(9,12)(10,19)(11,21)(16,29)(17,30)(18,28)(22,25)(23,26)(24,27)(31,32);(2,8)(3,13)(4,17)(7,19)(10,11)(12,26)(14,28)(15,31)(16,32)(20,21)(25,27)(29,30)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72240624, #72240640, #72251350, … |
| maximal normal subgroup of | #72240624, #72240640, #72251350, … |
| maximal quotient of | #72251442 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152478);