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 | |
11858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24);(5,6)(7,8)(9,11,14,21)(10,15,18,19)(12,17,22,24)(13,20,23,16);(2,4)(9,10,14,18)(11,19,21,15)(12,16,22,20)(13,17,23,24);(1,2)(3,4)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24);(9,12)(10,16)(11,17)(13,15)(14,22)(18,20)(19,23)(21,24);(9,13)(10,17)(11,20)(12,19)(14,23)(15,22)(16,21)(18,24)Polycyclic presentation:
2^9|b^2=i,c^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×#7684660, #72304537, … |
| maximal normal subgroup of | #72279669, C2×#7684660, #72304537, … |
| maximal quotient of | C2×#7684660, #72280735, #72281175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591856);