Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161854 |
| 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 | C2×C42 | C24 |
| prime radical | #35129 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1290 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)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(22,26,28,31)(27,33,34,36);(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,22,24,31,25,28,32,26)(23,27,30,36,29,34,35,33);(2,4)(7,15)(11,19)(14,20)(16,17)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(23,29)(27,34)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,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, 103762);