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 | |
2002 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)(6,12)(7,11)(8,16)(9,21)(13,29)(14,31)(15,28)(17,24)(18,32)(19,35)(20,26)(22,25)(27,33)(30,34);(5,6)(7,13)(8,18)(9,22)(10,15)(11,25)(12,21)(14,19)(16,34)(17,28)(20,31)(23,30)(24,29)(26,27)(32,35)(33,36);(2,4)(5,7,10,11)(6,13,15,25)(8,16,20,26)(9,17,24,21)(12,22,28,29)(14,30,32,33)(18,34,31,27)(19,23,35,36);(5,8,10,20)(6,14,15,32)(7,16,11,26)(9,23,24,36)(12,27,28,34)(13,30,25,33)(17,35,21,19)(18,29,31,22);(6,15)(12,28)(13,25)(14,32)(18,31)(22,29)(27,34)(30,33)Polycyclic presentation:
2^9|c^2=i,d^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 | #72286595, D4×#3414, #72309456, … |
| maximal normal subgroup of | #72286595, D4×#3414, #72309456, … |
| maximal quotient of | #72287259, #72309533, #72309991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530821);