Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,29,32,24)(20,26,33,42)(21,27,23,25)(30,43,44,38)(31,39,35,37)(34,40,36,41)(45,47,46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,35)(22,32)(23,31)(25,37)(26,38)(27,39)(28,29)(33,44)(34,46)(36,45)(40,47)(41,48)(42,43);(11,15)(13,16)(18,25)(20,33)(21,23)(24,37)(26,41)(27,28)(29,39)(30,44)(31,35)(38,48)(40,42)(43,47);(11,15)(13,16)(17,20)(18,26)(19,30)(21,36)(22,33)(23,34)(24,38)(25,40)(27,41)(28,42)(29,43)(31,46)(32,44)(35,45)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,22,23)(18,27,28,25)(19,31,32,35)(20,34,33,36)(24,37,29,39)(26,40,42,41)(30,45,44,46)(38,48,43,47)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6379440);