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 |
| prime radical | #59228 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(7,14)(10,16)(13,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,45)(36,42)(37,38)(39,47)(40,44)(41,48)(43,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,38,40,36)(34,37,44,42)(35,47,48,43)(39,41,46,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8079825);