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 | |
8363 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)(5,6,9,10)(7,11,12,8)(13,14,18,21)(15,19,23,17)(16,24,26,20)(22,25,28,27);(1,2)(3,4)(6,10)(7,12);(5,7)(6,11)(8,10)(9,12)(13,15,18,23)(14,17,21,19)(16,22,26,28)(20,25,24,27);(5,8)(6,12)(7,10)(9,11)(13,15)(14,19)(16,22)(17,21)(18,23)(20,27)(24,25)(26,28);(5,8)(6,7)(9,11)(10,12)(13,16,18,26)(14,20,21,24)(15,22,23,28)(17,25,19,27);(5,7)(6,11)(8,10)(9,12)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,24,28)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72279656, #72302931, #72302976, … |
| maximal normal subgroup of | #72279656, #72302931, #72302976, … |
| maximal quotient of | #72308965, #72308967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876288);