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 | |
11509 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)(9,10,14,13)(11,15,20,18)(12,16,22,21)(17,23,19,24);(5,6)(7,8)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,22)(21,23);(1,2,3,4)(9,12)(10,16)(11,17)(13,21)(14,22)(15,23)(18,24)(19,20);(1,2)(3,4)(9,13,14,10)(11,18,20,15)(12,21,22,16)(17,24,19,23);(9,12)(10,16)(11,19)(13,21)(14,22)(15,24)(17,20)(18,23);(9,11)(10,15)(12,17)(13,18)(14,20)(16,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=gi,c^2=h,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 | #72234472, #72234477, #72234484, … |
| maximal normal subgroup of | #72234472, #72234477, #72234484, … |
| maximal quotient of | #72261485, #72261491, #72261510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7598928);