Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162486 |
| 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 | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22,26,29,25,28,24,30)(23,31,32,27,34,36,33,35);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23)(22,27)(24,32)(25,34)(26,33)(28,35)(29,31)(30,36);(1,2,3,4)(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20)(22,28)(27,35)(29,30)(31,36);(7,14)(12,19)(15,18)(16,20)(21,24,25,26)(22,29,28,30)(23,32,34,33)(27,31,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112089);