Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6,8,10)(7,11,12,9)(14,19)(16,28)(17,31)(20,33)(22,35)(23,37)(24,40)(25,26)(29,34)(30,32)(39,43)(41,42);(1,2,3,4)(13,14,17,19,18,22,31,35)(15,23,25,37,27,33,26,20)(16,29,28,38,30,34,32,21)(24,36,41,43,42,44,40,39);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,26)(18,27)(19,33)(21,36)(22,37)(23,35)(25,31)(28,40)(29,39)(30,42)(32,41)(34,43)(38,44);(2,4)(13,16,18,30)(14,21,22,38)(15,24,27,42)(17,32,31,28)(19,34,35,29)(20,36,37,44)(23,39,33,43)(25,40,26,41)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332114);