Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,23,25,28)(16,22,18,21)(20,26,27,24);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(20,27)(21,22)(23,28);(2,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(1,2)(3,4)(13,16,17,18)(14,21,19,22)(15,24,25,26)(20,23,27,28);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72308515, #72308571, #72308641, … |
| maximal normal subgroup of | #72308515, #72308571, #72308641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481870);