Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,32)(21,25)(22,26)(23,31)(24,33)(27,35)(29,42)(30,38)(34,39)(36,48)(37,45)(40,47)(41,43)(44,46);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,36)(25,37)(26,28)(27,38)(32,46)(33,43)(34,44)(35,45)(39,42)(40,41)(47,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,33,22,34,35,47)(18,24,27,32,26,40,25,39)(19,29,30,43,31,44,45,48)(23,36,38,46,28,41,37,42);(11,15)(13,16)(17,21,22,35)(18,25,26,27)(19,30,31,45)(20,33,34,47)(23,37,28,38)(24,39,40,32)(29,43,44,48)(36,42,41,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,31)(23,28)(29,44)(30,45)(36,41)(37,38)(42,46)(43,48)Polycyclic presentation:
2^9|c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382034);