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 | |
11900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,24)(20,23);(5,6)(7,8)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24);(2,4)(10,16)(11,12)(18,21)(19,20);(1,2)(3,4)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)Polycyclic presentation:
2^9|b^2=i,d^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72241070, #72265952, #72265953, … |
| maximal normal subgroup of | #72241070, #72265952, #72265953, … |
| maximal quotient of | #72241115, #72268519, #72268523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578473);