Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35078 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(21,22)(23,26,30,33)(24,27)(25,28)(29,32,35,36)(31,34);(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,23,25,30)(22,26)(24,29,31,35)(27,32)(28,33)(34,36);(1,2,3,4)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(7,15)(11,19)(14,20)(16,17)(23,30)(24,31)(26,33)(27,34)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112234);