Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,13)(12,14)(15,16)(18,19)(20,24);(5,6,7,8)(10,11)(14,15)(18,19)(22,23);(5,6)(7,8)(12,16)(14,15)(17,18,21,19)(20,22,24,23);(1,2,3,4)(9,11,13,10)(12,14,16,15)(17,19,21,18)(20,23,24,22);(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72261384, #72261419, #72261553, … |
| maximal normal subgroup of | #72261384, #72261419, #72261553, … |
| maximal quotient of | #72278726, #72279050, C2×#3464941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3372137);