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 | |
11293 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)(3,4)(5,6,7,8)(9,10,15,12)(11,16,19,20)(13,17,23,21)(14,18,24,22);(5,6)(7,8)(9,11)(10,16)(12,20)(13,18)(14,21)(15,19)(17,24)(22,23);(2,4)(10,12)(11,19)(17,21)(18,22);(1,2)(3,4)(9,12,15,10)(11,20,19,16)(13,21,23,17)(14,22,24,18);(9,13)(10,17)(11,18)(12,21)(14,20)(15,23)(16,24)(19,22);(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,19)(20,23)Polycyclic presentation:
2^9|a^2=gi,d^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72234480, #72234484, #72234485, … |
| maximal normal subgroup of | #72234480, #72234484, #72234485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7598748);