Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×B(4,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1141 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)(7,15)(8,19)(9,13)(10,14)(11,17)(12,21)(16,30)(18,28)(20,34)(22,27)(23,25)(24,36)(26,35)(29,32)(31,33);(1,2,4,3)(5,7)(6,11)(8,16)(9,17)(10,23)(12,24)(13,15)(14,28)(18,22)(19,33)(20,31)(21,32)(25,27)(26,29)(30,34)(35,36);(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,26)(14,19)(15,29)(17,31)(18,32)(22,35)(23,36)(25,33)(27,34)(28,30);(1,3,4,2)(6,13)(7,17)(12,26)(14,27)(16,31)(18,23)(19,34)(32,36)Polycyclic presentation:
2^9|b^2=ei,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312326, #72312351, #72312355, … |
| maximal normal subgroup of | #72312326, #72312351, #72312355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408020);