Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,19,24);(5,6,7,8)(9,11)(10,13)(12,19)(14,17)(15,22)(16,24)(18,20)(21,23);(2,3)(10,15)(11,17)(12,20)(21,24);(1,3,4,2)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,20)(23,24);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,19)(22,24);(9,13)(10,11)(12,21)(14,22)(15,17)(16,18)(19,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=hi,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 | #72262026, #72262059, #72262101, … |
| maximal normal subgroup of | #72262026, #72262059, #72262101, … |
| maximal quotient of | #72297223, #72297296, #72297341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7583424);