Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129565 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12,15,28)(7,17,19,35)(8,21)(9,22)(10,24)(13,30)(14,25,29,16)(18,26,23,20)(27,36)(32,33);(1,2)(3,4)(5,6)(7,13)(8,14)(9,12)(10,25)(11,15)(16,24)(17,34)(18,36)(19,33)(20,30)(21,28)(22,29)(23,27)(26,32)(31,35);(2,4)(5,7)(6,13)(8,17)(9,23)(10,26)(11,19)(12,27)(14,34)(15,33)(16,30)(18,21)(20,24)(22,35)(25,32)(28,36)(29,31)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250029, #72250041, #72250044, … |
| maximal normal subgroup of | #72250029, #72250041, #72250044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55089);