Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1826 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)(7,10)(8,19)(9,23,26,18)(11,15)(12,21)(13,24,29,25)(14,16)(17,22)(20,33,36,27)(28,30)(31,35,32,34);(1,2,3,4)(5,7,16,15,11,14,10,6)(8,17,30,28,22,19,21,12)(9,18,24,29,26,23,25,13)(20,31,34,33,36,32,35,27);(5,8,16,30,11,22,10,21)(6,12,7,17,15,28,14,19)(9,20,24,34,26,36,25,35)(13,27,18,31,29,33,23,32);(5,9)(6,13)(7,18)(8,20)(10,25)(11,26)(12,27)(14,23)(15,29)(16,24)(17,31)(19,32)(21,35)(22,36)(28,33)(30,34)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312238, #72312252, #72312265 |
| maximal normal subgroup of | #72312238, #72312252, #72312265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300288);