Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454986 |
| 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×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 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)(5,6,10,15,11,16,9,14)(7,17,20,12,21,28,19,29)(8,22,24,33,25,32,26,13)(18,30,35,31,36,27,23,34);(1,3,4,2)(5,7)(6,12)(8,18,25,36)(9,20)(10,19)(11,21)(13,30,33,27)(14,28)(15,17)(16,29)(22,34,32,31)(23,26,35,24);(2,3)(5,8,9,24,11,25,10,26)(6,13,14,22,16,33,15,32)(7,18,19,35,21,36,20,23)(12,27,17,34,29,30,28,31);(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,31)(14,28)(15,17)(16,29)(18,26)(22,30)(24,36)(25,35)(27,32)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,e^2=i,f^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,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,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, 364084);