Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1959 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)(5,6,11,16)(7,17,20,25)(8,14,23,29)(9,12,10,15)(13,24,28,19)(18,32,34,31)(21,30,22,26)(27,35,36,33);(6,12)(8,21)(9,10)(13,17)(14,29)(15,16)(18,33)(19,24)(22,23)(25,28)(31,32)(34,35);(5,7)(6,13)(8,18)(9,19)(10,24)(11,20)(12,25)(14,27)(15,17)(16,28)(21,33)(22,35)(23,34)(26,32)(29,36)(30,31);(5,8,11,23)(6,14,16,29)(7,18,20,34)(9,22,10,21)(12,26,15,30)(13,27,28,36)(17,31,25,32)(19,35,24,33)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255707, #72256282, #72256289 |
| maximal normal subgroup of | #72255707, #72256282, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290361);