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 | |
4062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,14)(5,10,11,6)(12,16,15,13)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(33,34)(35,38)(36,45)(37,41)(39,42)(40,44)(43,48)(46,47);(1,3,7,4)(2,8,9,14)(5,12,11,15)(6,13,10,16)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(33,35)(34,38)(36,37)(39,43)(40,46)(41,45)(42,48)(44,47);(2,9)(3,4)(6,10)(12,15)(18,24)(21,30)(23,26)(27,28)(35,43)(36,44)(38,48)(40,45);(1,4,7,3)(2,8,9,14)(5,12,11,15)(6,16,10,13)(17,18)(19,23)(20,26)(21,25)(22,24)(27,31)(28,32)(29,30)(33,36)(34,40)(35,37)(38,46)(39,44)(41,48)(42,45)(43,47);(1,5,7,11)(2,10,9,6)(3,12,4,15)(8,16,14,13)(17,20,22,19)(18,23,24,26)(21,28,30,27)(25,31,29,32)(33,37)(34,41)(35,44)(36,43)(38,40)(39,47)(42,46)(45,48);(1,6,7,10)(2,11,9,5)(3,13,4,16)(8,15,14,12)(17,21,22,30)(18,25,24,29)(19,27,20,28)(23,31,26,32)(33,38)(34,35)(36,46)(37,40)(39,48)(41,44)(42,43)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835810);