Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,32)(18,22)(19,27)(20,26)(23,35)(25,39)(28,44)(29,38)(30,42)(31,40)(33,37)(34,41)(36,43);(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,37)(16,28,31,43)(17,33,35,27)(20,32,24,23)(21,38,40,39)(25,34,29,30)(36,42,44,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,29)(17,27)(18,26)(19,23)(21,39)(22,24)(25,31)(28,34)(30,43)(32,37)(33,35)(36,41)(38,40)(42,44);(7,12)(9,11)(13,16,18,31)(14,21,22,40)(15,25,26,29)(17,34,35,30)(19,36,37,44)(20,38,24,39)(23,42,32,41)(27,43,33,28);(6,10)(9,11)(14,22)(15,26)(21,40)(23,32)(25,29)(27,33)(28,43)(41,42)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901153);