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 | |
7681 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)(6,10)(8,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,26)(27,28);(1,2,3,4)(6,10)(7,11)(13,15)(14,19)(16,17)(18,24)(20,28)(21,25)(22,23)(26,27);(5,6,9,10)(7,8,11,12)(15,24)(17,27)(19,23)(21,28);(6,10)(7,11)(13,16)(14,20)(15,17)(18,26)(19,28)(21,23)(22,25)(24,27);(5,7)(6,8)(9,11)(10,12)(13,14)(15,19)(16,25)(17,21)(18,22)(20,26)(23,24)(27,28);(5,8)(6,7)(9,12)(10,11)(13,17)(14,21)(15,16)(18,27)(19,25)(20,23)(22,28)(24,26)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238143, #72238147, #72238151, … |
| maximal normal subgroup of | #72238143, #72238147, #72238151, … |
| maximal quotient of | #72300424, #72300738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7904515);