Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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×#3283 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,19)(12,16)(13,26)(14,17)(15,23)(18,27)(20,33)(21,30)(22,29)(24,37)(25,31)(28,36)(32,38)(34,40)(35,39);(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,30)(18,29)(19,26)(23,27)(24,34)(25,35)(31,40)(32,39)(33,37)(36,38);(2,4)(9,12,14,25)(10,16,17,31)(11,20,22,35)(13,24,27,38)(15,28,30,40)(18,32,26,37)(19,33,29,39)(21,34,23,36);(11,21)(13,27)(15,29)(18,26)(19,30)(20,34)(22,23)(24,38)(28,39)(32,37)(33,40)(35,36);(5,7,8,6)(10,17)(11,22)(16,31)(18,26)(20,35)(21,23)(32,37)(34,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72343336, #72343372, #72343397, … |
| maximal normal subgroup of | #72343336, #72343372, #72343397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291155);