Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C27 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
10174 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 10174 conjugacy classes of subgroups — too many to draw. Only its 650 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(14,18)(15,19)(21,22,24,26)(23,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,25)(24,26)(27,28);(7,11)(10,12)(14,18)(16,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219200, #72219202, #72219811, … |
| maximal normal subgroup of | #72219200, #72219202, #72219811, … |
| maximal quotient of | #72219195, #72225667, #72227779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4321325);