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 | |
7004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,10)(8,12)(13,14)(15,16)(17,25)(18,22)(19,20)(21,27)(23,28)(24,26);(1,2,3,4)(5,6,9,10)(7,8,11,12)(13,15,19,18)(14,16,20,22)(17,23,27,26)(21,24,25,28);(6,10)(7,11)(14,20)(16,22)(21,25)(24,28);(13,16,19,22)(14,15,20,18)(17,24,27,28)(21,23,25,26);(5,7,9,11)(6,8,10,12)(13,17,19,27)(14,21,20,25)(15,23,18,26)(16,24,22,28);(5,8)(6,11)(7,10)(9,12)(13,18,19,15)(14,22,20,16)(17,26,27,23)(21,28,25,24)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618326);