Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,9,4,8)(5,6,11,10)(13,14,15,16)(20,28)(24,29)(27,32)(30,31)(33,34,39,37)(35,42,40,36)(38,41,47,45)(43,46,48,44);(1,3,7,4)(2,8,12,9)(5,15,11,13)(6,14,10,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(34,37)(36,42)(41,45)(44,46);(1,3,7,4)(2,9,12,8)(5,13,11,15)(6,14,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,4,7,3)(2,9,12,8)(5,15,11,13)(6,14,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,5,7,11)(2,10,12,6)(3,13,4,15)(8,14,9,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,6,7,10)(2,11,12,5)(3,14,4,16)(8,13,9,15)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8403762);