Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1887 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)(13,14,18,22)(15,24,25,20)(16,21,31,38)(17,19,32,23)(26,37,29,36)(27,35,40,33)(28,34,30,39)(41,43,42,44);(5,6)(7,11)(8,10)(9,12)(14,19)(15,25)(16,28)(17,32)(20,35)(22,23)(24,33)(26,41)(29,42)(30,31)(34,39)(36,37);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,27)(18,25)(19,33)(21,37)(22,24)(23,35)(26,31)(28,41)(30,42)(32,40)(34,44)(36,38)(39,43);(7,12)(9,11)(13,16,18,31)(14,21,22,38)(15,26,25,29)(17,28,32,30)(19,34,23,39)(20,36,24,37)(27,42,40,41)(33,43,35,44);(6,10)(9,11)(14,22)(15,25)(19,23)(21,38)(26,29)(27,40)(34,39)(41,42)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970129);