Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| 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×#2200 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1158 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)(7,11)(8,18)(9,14)(10,13)(12,19)(15,29)(16,25)(17,24)(20,32)(21,28)(22,27)(23,30)(26,33)(31,36)(34,35);(1,2,4,3)(5,7,10,17)(6,11,13,24)(8,15,20,31)(9,21,22,16)(12,23,26,35)(14,28,27,25)(18,29,32,36)(19,30,33,34);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,26)(14,18)(16,30)(17,31)(21,34)(22,33)(24,35)(25,29)(27,32)(28,36);(1,3,4,2)(6,13)(11,24)(12,26)(14,27)(18,32)(23,35)(25,28)(29,36)Polycyclic presentation:
2^9|b^2=ei,d^2=e,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312506, #72312509, #72312517, … |
| maximal normal subgroup of | #72312506, #72312509, #72312517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407960);