Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459487 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2679 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1362 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)(7,15)(8,19)(9,23)(10,14)(11,18)(12,22)(13,24)(16,32)(17,31)(20,28)(21,29)(25,33)(26,35)(27,36)(30,34);(5,7)(6,11,14,26)(8,16)(9,18)(10,17)(12,27,19,25)(13,15,23,31)(20,36)(21,33)(22,34)(24,35)(28,32,29,30);(5,8,9,20,10,22,24,21)(6,12,13,28,14,19,23,29)(7,16,18,36,17,34,35,33)(11,25,15,30,26,27,31,32);(2,4)(7,17)(8,20,22,21)(11,26)(12,28,19,29)(15,31)(16,33,34,36)(18,35)(25,32,27,30)Polycyclic presentation:
2^9|b^2=g,c^2=ef,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279916 |
| maximal normal subgroup of | #72279916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373921);