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 | |
9014 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)(7,8)(9,10)(11,12)(14,19)(15,21)(20,26)(22,27);(1,2)(3,4)(13,14)(15,17)(16,20)(18,19)(21,25)(22,24)(23,26)(27,28);(6,10)(7,11)(14,19)(15,21)(16,23)(24,28);(5,7,9,11)(6,8,10,12)(13,15,18,21)(14,17,19,25)(16,22,23,27)(20,24,26,28);(5,8)(6,11)(7,10)(9,12)(13,16,18,23)(14,20,19,26)(15,22,21,27)(17,24,25,28);(13,17)(14,21)(15,19)(16,24)(18,25)(20,27)(22,26)(23,28)Polycyclic presentation:
2^9|d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238157, #72238215, #72238230, … |
| maximal normal subgroup of | #72238157, #72238215, #72238230, … |
| maximal quotient of | #72258893, #72258897, #72263374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650391);