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 | |
9322 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)(6,9)(7,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28);(1,2)(3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(14,18)(19,21)(20,24)(26,27);(5,7)(6,10)(8,11)(9,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)Polycyclic presentation:
2^9|d^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72256661, #72256690, #72256693, … |
| maximal normal subgroup of | #72256661, #72256690, #72256693, … |
| maximal quotient of | #72310680, #72327939, #72327942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549845);