Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1633 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12)(3,9,4,14)(5,11,6,10)(8,15,16,13)(17,18,22,26,23,19,21,25)(20,27,29,31,30,24,28,32);(1,3,7,4)(2,8,12,16)(5,13,6,15)(9,11,14,10)(17,19,23,18)(20,27,30,24)(21,26,22,25)(28,32,29,31);(1,4,7,3)(2,9,12,14)(5,13,6,15)(8,11,16,10)(17,20,22,29,23,30,21,28)(18,24,26,32,19,27,25,31);(1,5,7,6)(2,10,12,11)(3,13,4,15)(8,9,16,14)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32);(18,19)(24,27)(25,26)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dhi,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286982, #72287094, #72287142, … |
| maximal normal subgroup of | #72286982, #72287094, #72287142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252920);