Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2085 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,14,10,12,22,30)(7,15,19,28,17,11,20,27)(8,13,23,32,24,29,21,36)(16,25,18,33,34,31,35,26);(1,2,4,3)(5,7)(6,11,12,15)(8,16)(9,19)(10,17)(13,31,29,25)(14,27,30,28)(18,23)(20,22)(21,35)(24,34)(26,36,33,32);(1,3,4,2)(6,12)(7,16,17,34)(8,21)(9,22)(11,25,15,31)(13,32)(18,19,35,20)(23,24)(26,28,33,27)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=di,f^b=fh,h^b=hi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240845, #72240846, #72240850, … |
| maximal normal subgroup of | #72240845, #72240846, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56149);