Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,14,11,16,9,15)(7,17,20,27,21,31,19,12)(8,13,25,30,23,29,24,32)(18,33,35,36,34,28,22,26);(5,7,9,19,11,21,10,20)(6,12,14,17,16,27,15,31)(8,22,24,34,23,35,25,18)(13,28,30,26,29,33,32,36);(1,2,3,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,23)(12,26)(14,30)(15,32)(16,29)(17,33)(19,22)(20,35)(21,34)(27,36)(28,31);(8,23)(13,29)(18,34)(22,35)(24,25)(26,36)(28,33)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250755, #72250763, #72250791, … |
| maximal normal subgroup of | #72250755, #72250763, #72250791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265685);