Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(9,10)(13,17)(14,30)(15,16)(19,20)(22,34)(23,25)(26,32)(27,36)(28,29)(31,35);(1,2)(3,4)(5,6,9,12,11,16,10,15)(7,17,19,13,21,28,20,29)(8,14,25,26,24,32,23,30)(18,31,34,27,33,36,22,35);(2,4)(5,7,11,21)(6,13,16,29)(8,22,24,34)(9,20,10,19)(12,17,15,28)(14,31,32,36)(18,25,33,23)(26,35,30,27);(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,26)(13,27)(15,30)(16,32)(17,31)(19,34)(20,22)(21,33)(28,36)(29,35)Polycyclic presentation:
2^9|b^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72318890, #72318935, #72318947 |
| maximal normal subgroup of | #72318890, #72318935, #72318947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362888);