Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,20)(9,12)(10,16)(11,15)(14,22)(17,27)(18,32)(19,25)(21,30)(23,29)(24,26)(28,33)(31,35)(34,36);(6,12)(7,17,19,23)(13,27,25,29)(14,26)(15,16)(18,31,34,33)(20,30)(28,32,35,36);(2,4)(5,7)(6,13)(8,18)(9,19)(10,23)(11,17)(12,25)(14,28)(15,27)(16,29)(20,32)(21,34)(22,33)(24,31)(26,35)(30,36);(5,8)(6,14)(7,18)(9,21)(10,22)(11,24)(12,26)(13,28)(15,20)(16,30)(17,31)(19,34)(23,33)(25,35)(27,32)(29,36)Polycyclic presentation:
2^9|b^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239618, C2×#394354, #72341487, … |
| maximal normal subgroup of | #72239618, C2×#394354, #72341487, … |
| maximal quotient of | #72249691, C2×#394354, #72316663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301550);