Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 | #23920 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2302 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(6,9)(7,11)(13,15,18,25)(14,19,22,28)(16,23,17,24)(20,26,21,27);(2,4)(14,20)(16,17)(19,26)(21,22)(23,24)(27,28);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,22,20)(15,23,25,24)(19,27,28,26)Polycyclic presentation:
2^9|b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72307963, #72307986, #72308004 |
| maximal normal subgroup of | #72307963, #72307986, #72308004 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580063);