Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499780 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2958 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)(13,14,18,21)(15,19,23,28)(16,22,17,20)(24,26,25,27);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,28)(22,26);(5,7)(6,9)(8,12)(10,11)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(7,12)(9,11)(13,16,18,17)(14,20,21,22)(15,24,23,25)(19,27,28,26);(6,10)(9,11)(14,21)(15,23)(20,22)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281826, #72282331, #72282816 |
| maximal normal subgroup of | #72281826, #72282331, #72282816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407537);