Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
700 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)(5,6,9,15,10,12,25,26,11,16,23,33,8,14,19,29)(7,17,21,34,18,28,35,32,22,30,24,36,20,13,27,31);(2,4)(6,12,16,14)(7,18,22,20)(9,23)(13,28)(15,29,33,26)(17,30)(19,25)(21,27,24,35);(5,7,19,35,8,20,23,21,11,22,25,27,10,18,9,24)(6,13,29,34,14,30,33,31,16,28,26,36,12,17,15,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,f^b=fi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239414, #72239418, #72239430, … |
| maximal normal subgroup of | #72239414, #72239418, #72239430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54569);