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 | |
8392 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)(5,6,10,12)(7,9,8,11)(17,25)(21,26)(22,27)(24,28);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,14,18,19)(15,16,20,23)(17,21,25,26)(22,24,27,28);(6,12)(7,8)(14,19)(16,23)(21,26)(24,28);(5,6,10,12)(7,11,8,9)(13,15,18,20)(14,16,19,23)(17,22,25,27)(21,24,26,28);(5,8,10,7)(6,9,12,11)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,26)(25,28);(5,9)(6,7)(8,12)(10,11)(13,17)(14,21)(15,22)(16,24)(18,25)(19,26)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304554, #72304570, #72304601, … |
| maximal normal subgroup of | #72304554, #72304570, #72304601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617703);