Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,12,18)(7,19,22,36)(8,23,25,31)(9,15,28,35)(10,16,11,17)(13,20,30,21)(14,29,32,24)(26,33,27,34);(1,2)(3,4)(5,7)(6,13)(8,9)(10,20)(11,21)(12,22)(14,15)(16,19)(17,36)(18,30)(23,34)(24,26)(25,28)(27,29)(31,33)(32,35);(2,4)(5,8)(6,14)(7,9)(10,24)(11,29)(12,25)(13,15)(16,31)(17,23)(18,32)(19,33)(20,26)(21,27)(22,28)(30,35)(34,36);(2,4)(5,9)(6,15)(7,8)(10,27)(11,26)(12,28)(13,14)(16,34)(17,33)(18,35)(19,23)(20,29)(21,24)(22,25)(30,32)(31,36);(5,10,12,11)(6,16,18,17)(7,20,22,21)(8,24,25,29)(9,26,28,27)(13,19,30,36)(14,31,32,23)(15,33,35,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72311254, #72311366, #72311454, … |
| maximal normal subgroup of | #72311254, #72311366, #72311454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344677);