Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,20)(15,21,26,32)(16,22,28,33)(17,29,19,23)(24,34,41,42)(25,36,30,35)(27,38,31,37)(39,44,40,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,20)(16,27)(17,19)(21,32)(22,37)(24,39)(25,30)(28,31)(33,38)(34,43)(40,41)(42,44);(2,4)(13,15,18,26)(14,21,20,32)(16,24,28,41)(17,30,19,25)(22,34,33,42)(23,36,29,35)(27,39,31,40)(37,43,38,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,24)(17,27)(18,28)(19,31)(20,33)(21,34)(23,37)(25,40)(26,41)(29,38)(30,39)(32,42)(35,44)(36,43);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,23,20,29)(15,25,26,30)(16,27,28,31)(21,35,32,36)(22,37,33,38)(24,40,41,39)(34,44,42,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429657);