Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2482 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,25)(21,30)(22,26)(24,27)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,28)(18,23,26,31)(20,29,21,27)(24,25,32,30);(11,15)(13,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(2,6)(5,7)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,31,32);(3,8)(5,7)(11,15)(13,16)(19,28)(23,31)(24,32)(27,29)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282725, #72282746, #72282769, … |
| maximal normal subgroup of | #72282725, #72282746, #72282769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6652762);