Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,7,8)(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34)(35,36)(37,45)(38,39)(40,46)(41,44)(42,47)(43,48);(3,7)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(26,31)(34,39)(35,41)(37,46)(43,47);(1,3,5,7)(2,6,4,8)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,41)(34,36,39,44)(37,42,46,48)(40,43,45,47);(1,2,5,4)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,35)(37,47)(38,44)(39,41)(40,48)(42,45)(43,46);(1,4,5,2)(3,8,7,6)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=h,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3009378);