Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1379 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)(7,16,21,25)(8,13,23,29)(9,11)(10,15)(12,20,28,19)(14,17)(18,32)(22,30,24,26)(27,33)(31,35)(34,36);(6,11)(8,22)(9,17)(12,16)(13,29)(14,15)(18,33)(19,20)(23,24)(25,28)(31,32)(34,35);(2,4)(5,7,17,20,10,21,9,19)(6,12,14,16,15,28,11,25)(8,18,22,33,23,35,24,34)(13,27,30,31,29,36,26,32);(2,4)(5,8)(6,13)(7,18)(9,24)(10,23)(11,26)(12,27)(14,30)(15,29)(16,31)(17,22)(19,34)(20,33)(21,35)(25,32)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=egh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286356, #72287003, #72287067 |
| maximal normal subgroup of | #72286356, #72287003, #72287067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297608);