Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
4328 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)(7,10)(8,9)(11,12)(14,18)(15,20)(22,25)(23,27);(1,2)(3,4)(6,9)(10,12)(13,14)(15,16)(17,18)(19,20)(22,25)(26,28);(1,2,3,4)(7,11)(10,12)(14,18)(16,19)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,15,18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|c^2=hi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72262796, #72262824, #72262826, … |
| maximal normal subgroup of | #72262796, #72262824, #72262826, … |
| maximal quotient of | #72293943, C2×#2530461, #72318963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2437657);