Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,16)(14,19)(15,17)(18,20)(21,22,25,30,26,28,24,29)(23,31,33,27,34,36,32,35);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23,26,34)(22,27,28,35)(24,33,25,32)(29,36,30,31);(2,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(6,12)(7,14)(10,13)(15,20)(22,28)(27,35)(29,30)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92414);