Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(22,27)(24,25)(28,29)(30,34)(31,36)(32,35);(7,13)(8,16)(10,17)(11,15)(21,22,25,27,26,30,24,34)(23,31,32,36,33,28,35,29);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,33)(22,28,30,31)(24,35,25,32)(27,29,34,36);(7,13)(10,17)(14,19)(18,20)(21,23)(22,29)(24,32)(25,35)(26,33)(27,28)(30,36)(31,34);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,27,30,34)(23,32,33,35)(28,36,31,29)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282891, #72283184, #72283416, … |
| maximal normal subgroup of | #72282891, #72283184, #72283416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074231);