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 | |
8584 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)(5,6)(7,11)(8,9)(10,12)(15,22)(16,19)(23,27)(24,25);(1,2)(3,4)(13,14,18,21)(15,19,22,16)(17,20,26,28)(23,25,27,24);(6,9)(10,11)(13,15,18,22)(14,16,21,19)(17,23,26,27)(20,24,28,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,25)(24,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,16,18,19)(14,15,21,22)(17,25,26,24)(20,27,28,23);(13,17,18,26)(14,20,21,28)(15,23,22,27)(16,24,19,25)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72256676, #72256678, #72256681, … |
| maximal normal subgroup of | #72256676, #72256678, #72256681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7837707);