Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2387 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)(8,9)(11,12);(6,9)(10,12)(13,14)(15,22)(16,20)(17,27)(18,21)(19,25)(23,28)(24,26);(2,4)(13,15,18,24)(14,19,21,23)(16,26,27,22)(17,28,20,25);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,20,21,17)(15,23,24,19)(22,28,26,25);(14,16)(15,24)(17,20)(19,22)(21,27)(23,26)Polycyclic presentation:
2^9|c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307099, #72307116, #72307719 |
| maximal normal subgroup of | #72307099, #72307116, #72307719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482692);