Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×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:
(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,17,21,12,20,25,18,27)(8,13,24,30,22,28,23,29)(19,31,33,26,34,35,32,36);(7,18,20,21)(8,22)(12,25,27,17)(13,28)(19,33,34,32)(23,24)(26,31,36,35)(29,30);(2,4)(5,7)(6,12)(8,19)(9,21)(10,18)(11,20)(13,26)(14,25)(15,17)(16,27)(22,34)(23,33)(24,32)(28,36)(29,35)(30,31);(5,8,10,24,11,22,9,23)(6,13,15,30,16,28,14,29)(7,19,18,32,20,34,21,33)(12,26,17,31,27,36,25,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72252972, #72256910 |
| maximal normal subgroup of | #72231526, #72252972, #72256910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266400);