Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394378 |
| 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 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2558 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)(7,10)(8,12)(9,18)(11,15)(13,25)(14,16)(17,22)(19,31)(20,27)(21,28)(23,30)(24,29)(26,32)(33,36)(34,35);(5,7,16,15,11,14,10,6)(8,17,22,12,21,27,20,28)(9,18,25,30,24,29,23,13)(19,31,34,26,33,36,32,35);(2,4)(5,8)(6,12)(7,17)(9,19)(10,22)(11,21)(13,26)(14,27)(15,28)(16,20)(18,31)(23,34)(24,33)(25,32)(29,36)(30,35);(5,9,11,24)(6,13,15,30)(7,18,14,29)(8,19,21,33)(10,23,16,25)(12,26,28,35)(17,31,27,36)(20,32,22,34)Polycyclic presentation:
2^9|b^2=ei,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316837 |
| maximal normal subgroup of | #72316837 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301577);