Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,15)(12,14)(13,16)(17,20)(18,22)(19,21)(23,24);(5,6)(7,8)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(2,4)(6,8)(9,11,13,18)(10,15,16,22)(12,19,20,24)(14,21,17,23);(6,8)(11,18)(15,22)(19,24)(21,23);(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239649, #72249710, #72281266, … |
| maximal normal subgroup of | #72239649, #72249710, #72281266, … |
| maximal quotient of | #72239700, #72305535, C2×#6437416 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344612);