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 | |
7764 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)(13,14,18,20)(15,21,24,19)(16,25,26,17)(22,23,28,27);(1,2)(3,4)(13,15)(14,19)(16,22)(17,23)(18,24)(20,21)(25,27)(26,28);(6,9)(10,11)(13,15,18,24)(14,19,20,21)(16,22,26,28)(17,23,25,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,24)(20,21)(25,27)(26,28);(5,7,8,12)(6,10,9,11)(13,16,18,26)(14,17,20,25)(15,22,24,28)(19,23,21,27);(13,17)(14,16)(15,23)(18,25)(19,22)(20,26)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72236848, #72238150, #72238186, … |
| maximal normal subgroup of | #72236848, #72238150, #72238186, … |
| maximal quotient of | #72336620, #72336676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7823861);