Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #29607 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2924 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,25)(23,30,33,31)(24,34,29,26)(28,36,35,32);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,33)(22,28,25,35)(24,30,29,31)(26,36,34,32);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,26)(23,31)(25,34)(27,29)(28,32)(30,33)(35,36);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25,27,22)(23,28,33,35)(24,34,29,26)(30,36,31,32);(6,12)(10,17)(11,20)(18,19)(22,25)(24,29)(28,35)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388776);