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 | |
7482 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)(7,10)(8,12)(13,14,18,21)(15,19,24,27)(16,17,20,25)(22,23,26,28);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,15,18,24)(14,19,21,27)(16,22,20,26)(17,23,25,28);(5,7)(6,8)(9,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,24)(20,26)(21,27)(25,28);(5,6)(7,12)(8,10)(9,11);(5,8,9,12)(6,10,11,7)(13,16)(14,17)(15,22)(18,20)(19,23)(21,25)(24,26)(27,28);(5,6)(7,8)(9,11)(10,12)(13,17)(14,20)(15,23)(16,21)(18,25)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72280753 |
| maximal normal subgroup of | #72280753 |
| maximal quotient of | #72280809, #72280833, #72281695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7764913);