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 | |
11327 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)(11,15)(12,16)(13,17)(14,19)(18,20)(21,22)(23,24);(5,6,7,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,22)(17,24)(18,19);(1,2,3,4)(11,20)(15,18)(21,24)(22,23);(1,2)(3,4)(9,12)(10,16)(11,21)(13,19)(14,17)(15,22)(18,23)(20,24);(9,13,14,16)(10,17,19,12)(11,22,20,23)(15,21,18,24);(9,11)(10,18)(12,21)(13,22)(14,20)(15,19)(16,23)(17,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234475, #72234477, #72234485 |
| maximal normal subgroup of | #72234475, #72234477, #72234485 |
| maximal quotient of | #72261339, #72261346, #72261360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589948);