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 | |
8890 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)(5,6)(7,11)(8,9)(10,12);(1,2)(3,4)(13,14)(15,20)(16,21)(17,25)(18,19)(22,26)(23,24)(27,28);(6,9)(10,11)(14,19)(16,20)(22,25)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,20,19,16)(17,23,26,28)(22,27,25,24);(5,7)(6,10)(8,12)(9,11)(13,16,18,20)(14,21,19,15)(17,24,26,27)(22,28,25,23);(13,17)(14,22)(15,23)(16,24)(18,26)(19,25)(20,27)(21,28)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256612, #72256674, #72256676, … |
| maximal normal subgroup of | #72256612, #72256674, #72256676, … |
| maximal quotient of | #72263386, #72263390, #72263515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7635165);