Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2668 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,8)(5,9)(6,11)(7,15)(10,20)(13,28)(14,22)(16,26)(17,27)(18,24)(19,23)(21,29)(25,31)(30,32);(1,3)(2,7)(4,17)(5,14)(6,16)(8,21)(9,12)(10,26)(11,22)(13,18)(15,20)(19,30)(23,27)(24,32)(25,28)(29,31);(1,4)(2,8)(3,13)(5,19)(6,18)(7,21)(9,23)(10,25)(11,24)(12,27)(14,29)(15,30)(16,17)(20,31)(22,32)(26,28);(2,9)(3,14)(5,20)(8,23)(10,11)(12,26)(13,29)(15,16)(17,30)(19,31)(24,25)(27,28);(4,18)(8,24)(13,17)(19,31)(21,32)(23,25)(27,28)(29,30)Polycyclic presentation:
2^9|f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306892, #72306954, #72307052, … |
| maximal normal subgroup of | #72306892, #72306954, #72307052, … |
| maximal quotient of | #72238734, #72238735, #72238736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407944);