Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
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 | |
23874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(13,21)(17,22)(19,24)(20,23);(5,6)(7,8)(9,10)(11,16)(12,18)(13,17)(14,15)(19,23)(20,24)(21,22);(1,2)(3,4)(5,6)(7,8)(10,15)(11,18)(17,22)(19,24);(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20);(9,12,14,16)(10,11,15,18)(13,20,21,23)(17,19,22,24);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279275, #72284427, #72286845, … |
| maximal normal subgroup of | #72279275, #72284427, #72286845, … |
| maximal quotient of | #72306475, #72306510, #72308730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538349);