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 | |
7700 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)(13,14,18,17)(15,20,22,19)(16,24,26,21)(23,27,25,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,23)(17,20)(18,22)(21,27)(24,28)(25,26);(6,9)(10,12)(14,17)(15,22)(21,24)(23,25);(7,11)(10,12)(13,15)(14,20)(16,25)(17,19)(18,22)(21,27)(23,26)(24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,23)(17,24)(18,26)(19,27)(20,28)(22,25);(5,6)(7,10)(8,9)(11,12)(13,17,18,14)(15,19,22,20)(16,24,26,21)(23,27,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72333430 |
| maximal normal subgroup of | #72333430 |
| maximal quotient of | #72333448, #72333451, #72333481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7728425);