Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,23)(9,26)(10,13)(11,16)(12,20)(14,25)(15,24)(18,32)(19,30)(21,33)(22,28)(27,31)(29,36)(34,35);(1,2)(3,4)(5,7)(6,12)(8,19)(9,22)(10,21)(11,20)(13,28)(14,32)(15,30)(16,33)(17,26)(18,24)(23,34)(25,36)(27,35)(29,31);(2,4)(6,13)(12,28)(14,23)(15,31)(16,26)(17,33)(29,30)(32,34);(2,4)(5,8,10,25)(6,14,13,23)(7,18,21,35)(9,24,11,27)(12,29,28,30)(15,26,31,16)(17,34,33,32)(19,22,36,20);(5,8)(6,15)(7,19)(9,27)(10,25)(11,24)(12,30)(13,31)(14,16)(17,34)(18,20)(21,36)(22,35)(23,26)(28,29)(32,33)Polycyclic presentation:
2^9|d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,e^d=ei,h^d=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421386);