Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | #35102 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1423 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,26,29,32)(24,27)(25,28)(30,33,35,36)(31,34);(5,7,9,17,10,15,8,16)(6,11,12,14,13,19,18,20)(21,23)(22,26,28,32)(24,30)(25,29)(27,33,34,36)(31,35);(2,4)(7,15)(11,19)(14,20)(16,17)(23,29)(24,31)(26,32)(27,34);(5,8,10,9)(6,12,13,18)(7,16,15,17)(11,14,19,20)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,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, 103641);