Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9109 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4510 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)(5,6)(7,15,18,11)(8,12)(9,13)(10,14)(16,24,30,22)(17,28,31,23)(19,25)(20,26)(21,27)(29,34,36,33)(32,35);(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,23)(14,15)(19,29)(20,30)(21,31)(22,26)(25,34)(27,28)(32,36)(33,35);(2,4)(5,8)(6,12)(7,16)(9,19)(10,20)(11,22)(13,25)(14,26)(15,24)(17,29)(18,30)(21,32)(23,33)(27,35)(28,34)(31,36);(5,9,10,21)(6,13,14,27)(7,17,18,31)(8,19,20,32)(11,23,15,28)(12,25,26,35)(16,29,30,36)(22,33,24,34)Polycyclic presentation:
2^9|a^2=h,d^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237653, #72237655, #72237666, … |
| maximal normal subgroup of | #72237653, #72237655, #72237666, … |
| maximal quotient of | #72238119, #72251208, #72280921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264639);