Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,19)(20,24)(21,22);(9,11)(10,15)(12,20,22,17)(13,18)(14,19)(16,23,21,24);(2,4)(9,12)(10,16)(11,17)(13,21)(14,22)(15,23)(18,24)(19,20);(6,8)(9,13,14,10)(11,18,19,15)(12,21,22,16)(17,24,20,23)Polycyclic presentation:
2^9|b^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225689, #72234970, #72241081, … |
| maximal normal subgroup of | #72225689, #72234970, #72241081, … |
| maximal quotient of | #72225659, #72233893, #72241200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325019);