Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,16,24,15)(7,11,19,28,17,26,32,29)(8,20,22,13,23,12,21,30)(18,33,35,27,31,25,34,36);(1,2,3,4)(5,7)(6,11)(8,18)(9,19)(10,17)(12,25)(13,27)(14,28)(15,29)(16,26)(20,33)(21,34)(22,35)(23,31)(24,32)(30,36);(6,12)(8,21)(9,24)(11,25)(13,14)(15,30)(16,20)(18,34)(19,32)(22,23)(26,33)(27,28)(29,36)(31,35);(7,17)(11,26)(18,31)(19,32)(25,33)(27,36)(28,29)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239492, #72239528, #72239538, … |
| maximal normal subgroup of | #72239492, #72239528, #72239538, … |
| maximal quotient of | #72319185, #72320142, C2×#493777 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400973);