Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,14,15,16);(2,6)(7,8)(10,13)(14,16)(17,18)(19,22)(20,21)(23,24);(10,13)(14,16)(18,21)(22,24);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24);(19,23)(22,24);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72279634, #72280725, C2×#7629442, … |
| maximal normal subgroup of | #72279634, #72280725, C2×#7629442, … |
| maximal quotient of | #72280075, #72280376, #72280492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536638);