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 | |
8662 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)(7,11)(10,12)(13,14)(15,20)(16,18)(17,22)(19,21)(23,27)(24,26)(25,28);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,19,16)(14,20,21,18)(17,24,25,23)(22,26,28,27);(5,7)(6,10)(8,12)(9,11)(14,21)(15,16)(17,25)(26,27);(5,8,9,6)(7,10,11,12)(13,16,19,15)(14,18,21,20)(17,24,25,23)(22,26,28,27);(13,17)(14,22)(15,23)(16,24)(18,26)(19,25)(20,27)(21,28);(13,18)(14,15)(16,21)(17,26)(19,20)(22,23)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72261051, #72297209 |
| maximal normal subgroup of | #72261051, #72297209 |
| maximal quotient of | #72297234, #72297286, #72297543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581982);