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 | |
11500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(11,17)(13,21)(15,18)(16,20);(5,6)(7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24);(1,2)(3,4)(10,12)(15,18)(16,20)(22,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(2,4)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24);(1,2)(3,4)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,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, #72257416, #72312230, … |
| maximal normal subgroup of | D4×#3449, #72257416, #72312230, … |
| maximal quotient of | #72256929, #72264059, #72264109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572854);