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) = C26×C4 |
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 | |
3411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,19)(18,20)(21,22,24,25)(23,27,28,26);(1,2,3,4)(5,6,8,10)(7,9,12,11)(14,15)(16,20)(22,25)(23,28);(1,2)(3,4)(13,14,17,15)(16,18,20,19)(21,23)(22,26)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,28)(25,26);(5,6)(7,11)(8,10)(9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=i,d^2=fgi,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 | #72331971, #72331983, #72332050 |
| maximal normal subgroup of | #72331971, #72331983, #72332050 |
| maximal quotient of | #72333018 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2879891);