Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1094 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)(6,12)(8,22)(9,10)(13,17)(15,16)(18,33)(19,20)(23,27)(24,35)(25,26)(28,34)(29,30);(1,3,4,2)(5,6)(7,17)(8,23)(9,15)(10,12)(11,16)(13,20)(14,25)(18,34)(19,30)(21,29)(22,32)(24,31)(26,27)(28,35)(33,36);(2,3)(5,7,11,21)(6,13,16,30)(8,24,26,33)(9,20,10,19)(12,17,15,29)(14,31,32,36)(18,22,35,25)(23,28,27,34);(2,3)(5,8)(6,14)(7,18)(9,22)(10,25)(11,26)(12,27)(13,28)(15,23)(16,32)(17,31)(19,33)(20,24)(21,35)(29,36)(30,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384194);