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 | |
11246 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)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20);(5,6)(7,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24);(2,4)(10,16)(11,18)(17,23)(19,24);(1,2)(3,4)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24);(9,12,14,15)(10,11,16,18)(13,21,22,20)(17,24,23,19);(9,13,14,22)(10,17,16,23)(11,19,18,24)(12,20,15,21)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^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 | #72279669, C2×#7689162, #72304532, … |
| maximal normal subgroup of | #72279669, C2×#7689162, #72304532, … |
| maximal quotient of | C2×#7689162, #72280858, #72280940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7596358);