Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13404 |
| 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 | |
724 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)(5,6,10,13,11,17,9,15)(7,18,22,35,23,30,21,34)(8,24,26,14,27,33,25,16)(12,28,31,20,32,19,29,36);(5,7,8,20,10,21,26,28,11,23,27,36,9,22,25,19)(6,12,14,18,15,31,24,35,17,32,16,30,13,29,33,34);(2,4)(6,13)(7,19)(8,25)(9,10)(12,18)(14,16)(15,17)(20,22)(21,36)(23,28)(26,27)(29,35)(30,32)(31,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,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,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239848, #72239860, #72239861, … |
| maximal normal subgroup of | #72239848, #72239860, #72239861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55588);