Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,7,8)(9,10,14,12)(11,15,13,16)(18,22)(20,28)(23,25)(26,32)(33,34,38,35)(36,41,44,39)(37,40,46,42)(43,48,45,47);(3,7)(6,8)(9,11,14,13)(10,15,12,16)(20,28)(24,29)(26,32)(30,31)(34,35)(39,41)(40,42)(47,48);(1,3,5,7)(2,6,4,8)(9,12,14,10)(11,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,34)(36,41,44,39)(37,40,46,42)(43,47,45,48);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,39)(35,41)(37,45)(38,44)(40,48)(42,47)(43,46);(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,47)(41,48)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=h,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4078691);