Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162486 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22)(23,30,32,35)(24,33)(25,28)(26,29)(27,31,36,34);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23,25,34,26,32,24,31)(22,27,28,35,29,36,33,30);(2,4)(5,7,9,14)(6,12,13,19)(8,15,11,18)(10,16,17,20);(7,14)(12,19)(15,18)(16,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=i,e^2=hi,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, 103628);