Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(7,13)(8,16)(10,17)(12,18)(14,19)(15,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(2,4)(5,6)(7,14)(8,12)(9,11)(10,15)(13,19)(16,18)(17,20)(21,23,24,30,25,29,31,35)(22,26,27,33,28,32,34,36);(5,7,8,13,9,15,16,20)(6,10,12,17,11,14,18,19);(6,11)(10,14)(12,18)(17,19)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224967, #72225090, #72225120, … |
| maximal normal subgroup of | #72224967, #72225090, #72225120, … |
| maximal quotient of | #72224951, #72287325, C2×#203556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110752);