Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| 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 | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,14)(9,26)(10,18)(11,17)(12,16)(13,21)(15,24)(20,29)(22,33)(23,36)(25,32)(27,30)(28,34)(31,35);(1,2)(3,4)(5,7,12,27,11,22,10,21)(6,13,18,33,17,30,16,19)(8,23,24,35,9,20,25,34)(14,31,32,36,26,28,15,29);(2,4)(5,8,10,24,11,9,12,25)(6,14,16,32,17,26,18,15)(7,20,21,34,22,23,27,35)(13,28,19,29,30,31,33,36);(5,9)(6,15)(7,20)(8,11)(10,24)(12,25)(13,29)(14,18)(16,26)(17,32)(19,28)(21,35)(22,23)(27,34)(30,36)(31,33)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363158);