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 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(10,12)(13,14,16,17)(15,18,19,20)(22,27)(23,28)(24,25);(5,6)(7,10)(8,9)(11,12)(21,22)(23,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(21,23,24,28,26,22,25,27);(2,4)(6,9)(10,12)(13,15,16,19)(14,18,17,20)(21,23,25,27,26,22,24,28)Polycyclic presentation:
2^9|a^2=fg,c^2=e,d^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319974, #72320116, #72320325 |
| maximal normal subgroup of | #72319974, #72320116, #72320325 |
| maximal quotient of | #72321920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343394);