Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,27,20,29,19,12,17,28,14,11,7,10,15)(8,21,23,22,36,18,33,31,24,13,30,32,26,35,25,34);(1,2)(3,4)(5,7)(6,11)(8,22,24,32)(9,16)(10,19)(12,20)(13,30,21,23)(14,28)(15,29)(17,27)(18,25,35,33)(26,31,36,34);(2,4)(5,8,9,23,27,36,29,33,12,24,28,30,11,26,10,25)(6,13,14,22,20,35,15,31,17,21,16,32,7,18,19,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,f^b=fh,g^b=ghi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250454, #72250473, #72250485, … |
| maximal normal subgroup of | #72250454, #72250473, #72250485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56254);