Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2446 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)(16,21)(17,19)(18,22)(23,27)(24,28)(25,26);(5,6)(7,11)(8,10)(9,12)(14,19)(16,22)(17,21)(20,26)(23,28)(25,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,24)(14,20,22,28)(16,23,19,26)(17,25,21,27);(7,12)(9,11)(13,16,18,19)(14,21,22,17)(15,23,24,26)(20,27,28,25);(6,10)(9,11)(15,24)(20,28)(23,26)(25,27)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282743, #72283867, #72284118, … |
| maximal normal subgroup of | #72282743, #72283867, #72284118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901149);