Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2194 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)(5,6,10,15,12,16,11,27)(7,13,19,30,21,32,20,31)(8,22,24,33,9,25,26,14)(17,34,18,36,35,29,23,28);(1,2,3,4)(5,7)(6,13)(8,23)(9,18)(10,19)(11,20)(12,21)(14,29)(15,30)(16,32)(17,24)(22,28)(25,36)(26,35)(27,31)(33,34);(5,8,12,9)(6,14,16,33)(7,17,21,35)(10,26,11,24)(13,28,32,36)(15,25,27,22)(18,19,23,20)(29,31,34,30);(5,9,12,8)(6,14,16,33)(7,18,21,23)(10,26,11,24)(13,29,32,34)(15,22,27,25)(17,19,35,20)(28,31,36,30)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285177, #72285466, #72285507, … |
| maximal normal subgroup of | #72285177, #72285466, #72285507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302811);