Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1809 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,30)(25,31)(26,29)(27,32)(33,36)(34,35);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,33)(22,28,32,36)(24,25,34,26)(29,30,31,35);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,29)(23,33)(25,26)(28,35)(30,36)(31,32);(7,15)(10,13)(14,20)(17,19)(21,24)(22,30)(23,25)(26,33)(27,34)(28,31)(29,36)(32,35);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(21,25,27,26)(22,31,32,29)(23,24,33,34)(28,30,36,35)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306579);