Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1101 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)(5,6,11,14)(7,10,18,15)(8,12)(9,13)(16,22)(17,23)(19,32,34,26)(20,27)(21,28)(24,29)(25,30)(31,33,36,35);(5,7,15,14,11,18,10,6)(8,12,22,16,21,28,20,27)(9,17,23,30,25,29,24,13)(19,26,33,31,34,32,35,36);(5,8,15,22,11,21,10,20)(6,12,7,16,14,28,18,27)(9,19,23,33,25,34,24,35)(13,26,17,31,30,32,29,36);(2,4)(5,9)(6,13)(7,17)(8,19)(10,24)(11,25)(12,26)(14,30)(15,23)(16,31)(18,29)(20,35)(21,34)(22,33)(27,36)(28,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256961, #72257160, #72257207 |
| maximal normal subgroup of | #72256961, #72257160, #72257207 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309660);