Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(10,15)(12,13)(16,22)(17,21)(19,20)(23,24);(5,6)(7,8)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(2,4)(6,8)(9,11,14,18)(10,16,17,23)(12,19,13,20)(15,22,21,24);(5,6)(7,8)(11,18)(16,23)(19,20)(22,24);(6,8)(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72249710, #72281004, #72281013, … |
| maximal normal subgroup of | #72249710, #72281004, #72281013, … |
| maximal quotient of | #72249771, #72249777, #72249783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480984);