Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,15,16)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,36,41,44)(37,40,46,47)(42,43,48,45);(1,2,5,4)(3,6,7,8)(11,15)(14,16)(18,22)(19,25)(24,29)(26,31)(34,39)(35,41)(40,47)(42,48);(9,11,13,15)(10,14,12,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,35)(37,45)(38,44)(39,41)(40,48)(42,47)(43,46);(1,4,5,2)(3,8,7,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,47)(41,48)(44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=fh,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2624071);