Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13402 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
720 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)(5,6,11,14,12,18,9,16)(7,19,24,31,25,36,22,35)(8,26,27,15,28,17,10,29)(13,23,33,21,34,30,32,20);(5,7,8,21,11,22,27,23,12,25,28,20,9,24,10,30)(6,13,15,19,16,33,26,31,18,34,29,36,14,32,17,35);(2,4)(6,14)(7,20)(8,10)(9,11)(13,31)(15,29)(16,18)(19,33)(21,25)(22,23)(24,30)(27,28)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250226, #72250234, #72250282, … |
| maximal normal subgroup of | #72250226, #72250234, #72250282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55591);