Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,18)(17,20)(19,25)(21,26)(23,27)(24,28);(9,12)(14,18)(15,23)(16,26)(20,21)(24,25)(27,28);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,25)(14,19,20,22)(16,24,26,23)(18,27,21,28);(14,20)(18,21)(19,22)(27,28)Polycyclic presentation:
2^9|c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279295, #72279398 |
| maximal normal subgroup of | #72279295, #72279398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139413);