Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,30)(23,31,29,25)(24,34,27,28)(32,33,35,36);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,22)(23,32)(24,28)(25,33)(26,30)(27,34)(29,35)(31,36);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,23)(22,25)(24,33)(26,29)(27,36)(28,35)(30,31)(32,34);(6,12)(10,13)(11,18)(17,19)(21,24,26,27)(22,28,30,34)(23,33,29,36)(25,35,31,32);(7,15)(10,13)(14,20)(17,19)(21,25,26,31)(22,29,30,23)(24,32,27,35)(28,33,34,36)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576975);