Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,25)(19,27,29,23)(20,26,22,24)(28,32,30,31);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(11,15)(13,16)(18,24)(20,22)(23,31)(25,26)(27,32)(28,30);(2,6)(5,7)(18,25)(23,27)(24,26)(31,32);(3,8)(5,7)(11,15)(13,16)(17,20,21,22)(18,26,25,24)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72284379, #72285944, #72285987 |
| maximal normal subgroup of | #72284379, #72285944, #72285987 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6652843);