Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,11)(7,10,12,9)(14,17)(15,19)(21,22)(23,24)(25,28)(26,27);(5,7,8,12)(6,9,11,10)(13,14,16,17)(15,20,19,18)(21,22,25,28)(23,26,27,24);(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,20)(16,17)(18,19)(21,23)(22,24)(25,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72328475, #72328502, #72328563, … |
| maximal normal subgroup of | #72328475, #72328502, #72328563, … |
| maximal quotient of | #72330632, #72333049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2832006);