Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3057 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,28)(24,31)(25,30)(26,29)(27,33)(32,36)(34,35);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,26,32)(22,27,29,35)(24,33,25,34)(28,31,36,30);(2,4)(7,13)(10,17)(14,15)(18,19)(21,23)(22,28)(24,34)(25,33)(26,32)(27,30)(29,36)(31,35);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,29)(23,32)(30,31)(33,34);(6,12)(10,17)(11,20)(18,19)(21,24,26,25)(22,30,29,31)(23,33,32,34)(27,28,35,36)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388764);