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 | |
8256 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)(13,14,19,22)(15,16,17,20)(18,21,26,28)(23,24,25,27);(1,2)(3,4)(5,6)(7,12)(8,9)(10,11)(13,15,19,17)(14,20,22,16)(18,25,26,23)(21,24,28,27);(6,11)(7,8)(13,16,19,20)(14,15,22,17)(18,24,26,27)(21,23,28,25);(5,7,10,8)(6,12,11,9);(5,8,10,7)(6,9,11,12)(13,17,19,15)(14,16,22,20)(18,25,26,23)(21,24,28,27);(5,9)(6,7)(8,11)(10,12)(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284055, #72284359, #72304540, … |
| maximal normal subgroup of | #72284055, #72284359, #72304540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7644749);