Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,26,29)(23,27,25,34)(24,31,33,35)(28,32,36,30);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(21,23,26,25)(22,27,29,34)(24,32,33,30)(28,31,36,35);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,33)(22,28,29,36)(23,30,25,32)(27,35,34,31);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20);(1,3,4,2)(6,14)(11,15)(12,13)(19,20)(24,33)(28,36)(30,32)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307787);