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 | |
6314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,13)(14,16)(17,18,21,22)(19,20,24,23);(1,2,5,7)(3,4,8,6)(18,22)(20,23);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,23,22,20);(1,3,5,8)(2,6,7,4)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,23)(22,24);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340976 |
| maximal subgroup of | #72300811, #72326662, #72340976, … |
| maximal normal subgroup of | #72300811, #72326662, #72340976, … |
| maximal quotient of | #72300821, #72300935, #72326762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10273361);