Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11,14,27)(8,19)(9,23)(12,28,30,25)(13,26,15,20)(17,33)(18,32)(21,35)(22,24)(29,34,31,36);(5,6)(7,12)(8,20)(9,13)(10,14)(11,24)(15,23)(16,30)(17,34)(18,31)(19,27)(21,36)(22,26)(25,33)(28,35)(29,32);(2,4)(5,7)(6,12)(8,21)(9,18)(10,16)(11,25)(13,31)(14,30)(15,29)(17,22)(19,35)(20,36)(23,32)(24,33)(26,34)(27,28);(6,13,14,15)(7,16)(8,22)(11,26,27,20)(12,29,30,31)(18,32)(19,24)(25,36,28,34)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239796, #72239799, #72239843, … |
| maximal normal subgroup of | #72239796, #72239799, #72239843, … |
| maximal quotient of | #72265518, #72280614, #72285149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402929);