Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2719 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,22)(17,19)(18,21)(23,28)(24,26)(25,27);(5,7,9,6)(8,11,12,10)(14,19)(15,23)(16,18)(20,27)(24,25)(26,28);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,24)(14,20,19,26)(16,23,18,25)(21,27,22,28);(2,4)(6,7)(10,11)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,27,26,28);(13,16,17,18)(14,22,19,21)(15,23,24,25)(20,28,26,27)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281793, #72282335, #72282476, … |
| maximal normal subgroup of | #72281793, #72282335, #72282476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409041);