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 | |
7598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(6,10)(11,12)(13,14,19,21)(15,20,23,28)(16,25,26,18)(17,22,27,24);(5,6)(7,11)(8,12)(9,10)(13,15)(14,20)(16,17)(18,24)(19,23)(21,28)(22,25)(26,27);(2,4)(5,7,9,8)(6,11,10,12)(14,21)(15,23)(16,26)(22,24);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,18)(15,17)(19,26)(20,24)(21,25)(22,28)(23,27);(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,16)(18,28)(19,27)(20,25)(21,24)(23,26);(13,18)(14,16)(15,24)(17,20)(19,25)(21,26)(22,23)(27,28)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72238183, #72238192, #72238212, … |
| maximal normal subgroup of | #72238183, #72238192, #72238212, … |
| maximal quotient of | #72313247, #72313531, #72313734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603286);