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 | |
7094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(15,20)(17,22)(23,26)(25,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,16)(15,22,20,17)(18,21,27,24)(23,28,26,25);(6,9)(10,11)(13,15,19,20)(14,17,16,22)(18,26,27,23)(21,28,24,25);(7,12)(10,11)(13,16,19,14)(15,17,20,22)(18,24,27,21)(23,25,26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,16)(18,25)(19,22)(21,26)(23,24)(27,28);(5,6)(7,10)(8,9)(11,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7692530);