Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,20,27)(8,21)(9,14)(10,24)(11,15)(12,19,28,18)(13,23)(17,32,35,26)(22,30)(25,29)(31,33,36,34);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,26)(14,27)(15,28)(16,24)(21,31)(22,35)(23,34)(25,33)(29,32)(30,36);(5,8,9,23,11,22,10,25)(6,13,14,30,15,29,24,21)(7,17,18,34,20,35,19,33)(12,26,27,36,28,32,16,31);(7,18,20,19)(8,22)(12,27,28,16)(13,29)(17,33,35,34)(21,30)(23,25)(26,31,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi,d^c=di,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280083, #72280601 |
| maximal normal subgroup of | #72280083, #72280601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263949);