Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 |
| prime radical | #35222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(2,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(6,10)(7,11)(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(7,11)(9,12);(14,19)(18,25)(20,26)(24,28)Polycyclic presentation:
2^9|c^2=gh,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239690, #72240658, #72249430, … |
| maximal normal subgroup of | #72239690, #72240658, #72249430, … |
| maximal quotient of | #72308756, #72308969, C2×#6459347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366543);