Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,19,21)(15,24,27,28)(16,22,20,26)(17,23,18,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,22,19,27,17,26)(14,20,23,24,21,16,25,28);(2,4)(6,9)(10,12)(14,21)(16,20)(23,25)(24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,22)(15,25)(17,24)(18,28)(19,20)(21,26)(23,27);(7,11)(10,12)(13,17,19,18)(14,23,21,25)(15,26,27,22)(16,28,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264121, #72264140, #72264210 |
| maximal normal subgroup of | #72264121, #72264140, #72264210 |
| maximal quotient of | #72265684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452378);