Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
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 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,11)(9,25)(10,14)(12,18)(13,23)(15,22)(17,26)(19,31)(20,28)(21,29)(24,27)(30,34)(32,36)(33,35);(6,11,14,27)(8,20)(9,22)(12,29,16,26)(13,25,28,15)(17,34)(19,35)(21,36)(23,24)(30,33,32,31);(5,7)(6,12,14,16)(8,21)(9,19)(10,18)(11,26,27,29)(13,32,28,30)(15,31,25,33)(17,24)(20,36)(22,35)(23,34)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231475, #72231524, #72231546, … |
| maximal normal subgroup of | #72231475, #72231524, #72231546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56668);