Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,30)(24,34)(25,29)(26,28)(27,33)(31,32)(35,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,32)(25,33)(26,35)(28,31)(29,30)(34,36);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,22)(23,31)(24,28)(25,29)(26,34)(27,35)(30,32)(33,36);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,25,26)(22,28,29,34)(23,32,33,35)(27,31,30,36);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,32)(24,26)(27,36)(28,34)(30,31)(33,35)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281541, #72281978, #72282920, … |
| maximal normal subgroup of | #72281541, #72281978, #72282920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594607);