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 | |
11398 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)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23);(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23);(2,4)(10,15)(11,17)(18,23)(19,24);(1,2)(3,4)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)Polycyclic presentation:
2^9|d^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 | #72234472, #72234479, #72279669, … |
| maximal normal subgroup of | #72234472, #72234479, #72279669, … |
| maximal quotient of | C2×#7684861, #72280821, #72281099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592057);