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 | |
11644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,23)(20,21)(22,24);(5,6)(7,8)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)Polycyclic presentation:
2^9|a^2=g,c^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72297215, C2×#7631902 |
| maximal normal subgroup of | #72297215, C2×#7631902 |
| maximal quotient of | #72297238, #72297399, #72297614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7539098);