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 | |
11314 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);(5,6,7,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,2)(3,4)(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|b^2=g,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 | #72279669, C2×#7685071, #72304562, … |
| maximal normal subgroup of | #72279669, C2×#7685071, #72304562, … |
| maximal quotient of | C2×#7685071, #72280842, #72280858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592267);