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 | |
8122 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)(13,14)(15,19)(16,17)(18,26)(20,22)(21,27)(23,24)(25,28);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15,20,17)(14,19,22,16)(18,23,27,25)(21,28,26,24);(5,7)(6,9)(8,12)(10,11)(13,16,20,19)(14,15,22,17)(18,24,27,28)(21,23,26,25);(13,17,20,15)(14,16,22,19)(18,25,27,23)(21,24,26,28);(5,7)(6,10)(8,12)(9,11)(13,18)(14,21)(15,23)(16,24)(17,25)(19,28)(20,27)(22,26);(5,6)(7,9)(8,11)(10,12)(13,19,20,16)(14,15,22,17)(18,24,27,28)(21,25,26,23)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=h,f^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 | #72256671, #72256674, #72256697, … |
| maximal normal subgroup of | #72256671, #72256674, #72256697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7758695);