Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,25)(24,32)(26,30)(27,29)(28,34)(31,35)(33,36);(1,2,3,4)(6,9)(7,11)(13,15)(14,17)(16,19)(18,20)(21,23,27,30,26,29,25,22)(24,31,35,32,34,36,33,28);(6,9)(10,12)(15,19)(17,20)(21,24,26,34)(22,28,30,32)(23,31,29,36)(25,33,27,35);(2,4)(5,7,8,11)(6,10,9,12)(22,29)(23,30)(25,27)(28,36)(31,32)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343675);