Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(22,26)(24,27)(28,34)(29,30)(31,36)(33,35);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,22,27,29,25,30,24,26)(23,31,35,34,32,28,33,36);(2,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,23)(22,28)(24,33)(25,32)(26,34)(27,35)(29,36)(30,31);(8,15)(11,13)(16,20)(18,19)(22,29)(23,32)(24,27)(26,30)(28,34)(31,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281632, #72281954, #72282108, … |
| maximal normal subgroup of | #72281632, #72281954, #72282108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7098404);