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 | |
8900 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)(5,6,9,10)(7,8,11,12)(13,14)(15,23)(16,20)(17,18)(19,21)(22,28)(24,27)(25,26);(1,2)(3,4)(13,15,19,16)(14,20,21,23)(17,24,28,26)(18,25,22,27);(6,10)(7,11)(14,21)(15,16)(18,22)(24,26);(5,7)(6,8)(9,11)(10,12)(13,16,19,15)(14,20,21,23)(17,24,28,26)(18,27,22,25);(5,8)(6,7)(9,12)(10,11)(13,17)(14,22)(15,24)(16,26)(18,21)(19,28)(20,27)(23,25);(13,18,19,22)(14,17,21,28)(15,25,16,27)(20,24,23,26)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326920 |
| maximal normal subgroup of | #72326920 |
| maximal quotient of | #72327049, #72327989, #72329121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7691509);