Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13,15,11)(8,16)(9,10)(12,17)(14,18,20,19)(21,22)(23,31)(24,30)(25,27)(26,29)(28,32)(33,36)(34,35);(1,2,3,4)(6,10)(8,17)(11,13)(14,20)(22,27)(28,35)(29,30)(31,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dhi,f^c=fi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95729);