Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| 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 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1976 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)(5,6,11,8)(7,15,18,34)(9,22,26,33)(10,21,24,14)(12,29,16,19)(13,28,20,25)(17,35,27,31)(23,32,36,30);(5,7)(6,12)(8,16)(9,23)(10,27)(11,18)(13,31)(14,32)(15,22)(17,24)(19,25)(20,35)(21,30)(26,36)(28,29)(33,34);(2,4)(5,8,11,6)(7,16,18,12)(9,20,26,13)(10,21,24,14)(15,19,34,29)(17,32,27,30)(22,25,33,28)(23,35,36,31);(5,9)(6,13)(7,17)(8,20)(10,28)(11,26)(12,30)(14,33)(15,34)(16,32)(18,27)(19,29)(21,22)(24,25);(7,18)(12,16)(15,34)(17,27)(19,29)(23,36)(30,32)(31,35)Polycyclic presentation:
2^9|a^2=i,c^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286580, D4×#3414, #72309422, … |
| maximal normal subgroup of | #72286580, D4×#3414, #72309422, … |
| maximal quotient of | #72287072, #72309493, #72309977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530818);