Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,14)(9,17)(10,16)(11,15)(12,21)(13,23)(19,27)(20,34)(22,33)(24,32)(25,29)(26,30)(28,36)(31,35);(1,3,4,2)(5,7)(6,12)(8,19)(9,25)(10,21)(11,22)(13,27)(14,31)(15,29)(16,33)(17,18)(20,23)(24,36)(26,35)(28,30)(32,34);(5,8)(6,13)(7,19)(9,23)(10,26)(11,24)(12,27)(14,16)(15,30)(17,32)(18,34)(20,25)(21,35)(22,36)(28,29)(31,33);(5,8)(6,14)(7,20)(9,26)(10,23)(11,24)(12,28)(13,16)(15,32)(17,30)(18,34)(19,25)(21,36)(22,35)(27,29)(31,33);(6,15)(12,29)(13,30)(14,32)(16,17)(18,33)(27,28)(31,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,d^b=df,f^b=fi,g^c=gi,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72311981, #72312221 |
| maximal normal subgroup of | #72279230, #72311981, #72312221 |
| maximal quotient of | #72279546, #72309588, #72322077 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288668);