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 | |
8228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,18,21,26)(16,20,24,27)(22,25,23,28);(1,2,3,4)(13,15)(14,18)(16,23)(17,21)(19,26)(20,28)(22,24)(25,27);(6,9)(10,11)(14,19)(15,21)(20,27)(22,23);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,24)(27,28);(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,20)(17,19)(21,26)(22,25)(23,28)(24,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,27)(21,23)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72330777 |
| maximal normal subgroup of | #72330777 |
| maximal quotient of | #72330816, #72330818, #72332122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7633913);