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 | |
11389 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)(9,10,14,16)(11,15,19,21)(12,17,22,24)(13,18,23,20);(5,6,7,8)(9,11)(10,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24);(2,4)(10,16)(11,19)(17,24)(18,20);(1,2)(3,4)(9,11)(10,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24);(9,12)(10,17)(11,20)(13,15)(14,22)(16,24)(18,19)(21,23);(9,13)(10,18)(11,17)(12,21)(14,23)(15,22)(16,20)(19,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72279669, C2×#7689167, #72304534, … |
| maximal normal subgroup of | #72279669, C2×#7689167, #72304534, … |
| maximal quotient of | C2×#7689167, #72280694, #72280804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7596363);