Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| 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×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,14,11,16,10,15)(7,17,19,29,21,28,20,12)(8,22,24,32,26,31,25,13)(18,27,23,36,34,35,33,30);(1,3,4,2)(5,7)(6,12)(8,23,26,33)(9,20)(10,19)(11,21)(13,30,32,36)(14,28)(15,17)(16,29)(18,25,34,24)(22,35,31,27);(2,3)(5,8,9,25,11,26,10,24)(6,13,14,31,16,32,15,22)(7,18,19,33,21,34,20,23)(12,27,17,30,29,35,28,36);(6,14)(8,24)(9,10)(12,28)(15,16)(17,29)(18,23)(19,20)(22,31)(25,26)(27,35)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364078);