Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2109 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)(5,6)(7,10)(8,11)(9,12)(14,19)(20,22)(21,27)(23,28);(2,4)(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,23,25,28)(16,22,18,20)(21,26,27,24);(5,7,9,8)(6,11,12,10)(14,20)(16,18)(19,22)(21,23)(24,26)(27,28);(5,8,9,7)(6,11,12,10)(13,15)(14,21)(16,24)(17,25)(18,26)(19,27)(20,23)(22,28);(13,16,17,18)(14,22,19,20)(15,24,25,26)(21,28,27,23)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=g,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307412, #72307515, #72307607 |
| maximal normal subgroup of | #72307412, #72307515, #72307607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360306);