Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1945 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23,21,25,20,26)(19,24,29,30,27,31,28,32);(9,11)(10,13)(12,15)(14,16)(18,23)(20,22)(24,30)(25,26)(28,29)(31,32);(2,5)(6,8)(10,14)(11,15)(17,19,21,27)(18,24,25,31)(20,28,22,29)(23,30,26,32);(11,15)(13,16)(19,27)(24,31)(28,29)(30,32);(1,3,4,7)(2,6,5,8)(18,25)(23,26)(24,31)(30,32)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=gh,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263999, #72264113, #72264121, … |
| maximal normal subgroup of | #72263999, #72264113, #72264121, … |
| maximal quotient of | #72308511 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376027);