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 | |
12246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(6,8)(11,17)(12,19)(15,21)(16,20);(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(1,2)(3,4)(10,14)(15,21)(16,20)(22,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,23)(20,24);(1,2,3,4)(9,10)(11,15)(12,20)(13,14)(16,19)(17,21)(18,24)(22,23);(1,2)(3,4)(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#3449, #72267574, #72267577, … |
| maximal normal subgroup of | D4×#3449, #72267574, #72267577, … |
| maximal quotient of | #72257071, #72257099, #72257109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7571881);