Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
7556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(6,7)(10,15)(11,12)(17,18)(19,20)(21,23)(22,24);(1,2,5,3)(4,6,8,7)(9,10,14,15)(11,13,12,16)(19,24)(20,22);(9,11,14,12)(10,13,15,16)(17,19,21,24)(18,22,23,20);(1,2,5,3)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,20)(18,19)(21,22)(23,24);(1,3,5,2)(4,7,8,6);(1,4)(2,6)(3,7)(5,8)(9,13)(10,12)(11,15)(14,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232288, #72233860, #72241224, … |
| maximal normal subgroup of | #72232288, #72233860, #72241224, … |
| maximal quotient of | #72232348, #72232367, #72234034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345751);