Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1270 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)(5,6,11,15,12,8,10,16)(7,17,20,22,21,29,19,13)(9,14,26,31,27,23,25,32)(18,33,34,28,35,30,24,36);(5,7,10,19,12,21,11,20)(6,13,15,17,8,22,16,29)(9,24,25,35,27,34,26,18)(14,30,31,36,23,33,32,28);(1,3,4,2)(5,8,12,6)(7,17,21,29)(9,23,27,14)(10,16,11,15)(13,19,22,20)(18,33,35,30)(24,28,34,36)(25,32,26,31);(5,9,12,27)(6,14,8,23)(7,18,21,35)(10,26,11,25)(13,28,22,36)(15,32,16,31)(17,33,29,30)(19,34,20,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fhi,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285454, #72285571 |
| maximal normal subgroup of | #72284999, #72285454, #72285571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296326);