Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2024 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)(5,6,9,13)(7,15,17,34)(8,19,22,33)(10,24,21,14)(11,27,29,18)(12,26,31,23)(16,35,25,30)(20,32,36,28);(5,7)(6,11)(8,20)(9,17)(10,25)(12,30)(13,29)(14,32)(15,19)(16,21)(18,23)(22,36)(24,28)(26,27)(31,35)(33,34);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,28)(17,29)(19,23)(20,30)(21,24)(22,31)(25,32)(26,33)(27,34)(35,36);(5,8)(6,12)(7,16)(9,22)(10,26)(11,28)(13,31)(14,33)(15,34)(17,25)(18,27)(19,24)(21,23)(29,32);(6,13)(7,17)(12,31)(14,24)(16,25)(18,27)(19,33)(20,36)Polycyclic presentation:
2^9|a^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286697, #72315336, #72315837, … |
| maximal normal subgroup of | #72286697, #72315336, #72315837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530844);