Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
7007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,28)(25,26);(1,2)(3,4)(13,14,18,22)(15,20,16,19)(17,21,26,25)(23,28,24,27);(5,6,8,10)(7,9,12,11)(13,14)(15,19)(16,20)(17,25)(18,22)(21,26)(23,28)(24,27);(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,19,22,20)(17,23,26,24)(21,27,25,28);(5,7,8,12)(6,9,10,11)(13,16,18,15)(14,20,22,19)(17,24,26,23)(21,28,25,27);(13,17,18,26)(14,21,22,25)(15,23,16,24)(19,27,20,28)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670625);