Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,14,17)(11,12,15,19)(13,21,22,16)(18,24,23,20);(5,6)(7,8)(11,15)(12,19)(18,23)(20,24);(2,4)(9,11)(10,12)(13,18)(14,15)(16,20)(17,19)(21,24)(22,23);(1,2)(3,4);(9,12)(10,15)(11,17)(13,20)(14,19)(16,23)(18,21)(22,24);(9,13,14,22)(10,16,17,21)(11,18,15,23)(12,20,19,24)Polycyclic presentation:
2^9|a^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72280094, #72303001, #72303004, … |
| maximal normal subgroup of | #72280094, #72303001, #72303004, … |
| maximal quotient of | #72281269, #72281298 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7587357);