Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,13)(11,14,16,15)(20,27)(24,29)(25,31)(30,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,2,5,7)(3,6,8,4)(10,13)(14,15)(18,22)(23,28)(24,29)(30,32)(34,35)(39,41)(40,42)(46,47);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,14,13,15)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35,38,34)(36,39,44,41)(37,40,45,42)(43,47,48,46);(1,4)(2,3)(5,6)(7,8)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=fgh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5395624);