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 | |
11575 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)(6,8)(9,10)(11,20)(12,17)(13,18)(14,19)(15,16)(21,23)(22,24);(5,6,7,8)(9,11,15,19)(10,14,16,20)(12,13,21,24)(17,22,23,18);(2,4)(10,16)(11,19)(12,21)(18,22);(1,2)(3,4)(9,12,15,21)(10,17,16,23)(11,13,19,24)(14,22,20,18);(9,13)(10,18)(11,21)(12,19)(14,17)(15,24)(16,22)(20,23);(9,14)(10,19)(11,16)(12,22)(13,23)(15,20)(17,24)(18,21)Polycyclic presentation:
2^9|b^2=g,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 | #72238149, #72238219, #72238234, … |
| maximal normal subgroup of | #72238149, #72238219, #72238234, … |
| maximal quotient of | C2×#7684865, #72280784, #72280795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592061);