Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #25742 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,10)(7,12,16,25)(8,18,21,34)(9,22,24,14)(13,29,26,20)(15,31,32,35)(17,30,23,28)(19,33,27,36);(1,2,4,3)(5,7)(6,12)(8,19)(9,23)(10,25)(11,16)(13,15)(14,30)(17,24)(18,33)(20,35)(21,27)(22,28)(26,32)(29,31)(34,36);(5,8,9,20,11,21,24,29)(6,13,14,18,10,26,22,34)(7,15,17,33,16,32,23,36)(12,27,28,31,25,19,30,35);(1,3,4,2)(7,16)(12,25)(15,32)(17,23)(19,27)(28,30)(31,35)(33,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410415);