Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
4708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,13,12)(11,14,15,16)(17,18,22,21)(19,20,24,23);(1,2)(3,4)(6,8)(10,12)(14,16)(17,19,22,24)(18,23,21,20);(2,3)(5,6)(7,8)(10,12)(14,16)(18,21)(19,24);(2,3)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,24)(22,23);(1,3,4,2)(9,12,13,10)(11,14,15,16)(17,21,22,18)(19,23,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232560, #72232561, #72232568, … |
| maximal normal subgroup of | #72232560, #72232561, #72232568, … |
| maximal quotient of | #72330443, #72330726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3869545);