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 | |
4284 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,14)(15,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34,39,41)(35,42,46,37)(36,38,48,40)(43,47,44,45);(1,3,7,6)(2,8,11,12)(4,15,10,13)(5,16,9,14)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35)(34,37)(36,43)(38,45)(39,46)(40,47)(41,42)(44,48);(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,39,48)(34,40,41,38)(35,43,46,44)(37,47,42,45);(1,4,7,10)(2,9,11,5)(3,13,6,15)(8,16,12,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36)(34,38)(35,44)(37,47)(39,48)(40,41)(42,45)(43,46);(1,5,7,9)(2,10,11,4)(3,14,6,16)(8,15,12,13)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,37)(34,35)(36,47)(38,44)(39,42)(40,43)(41,46)(45,48);(1,6,7,3)(2,8,11,12)(4,15,10,13)(5,14,9,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38)(34,36)(35,45)(37,43)(39,40)(41,48)(42,44)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9843058);