Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3176 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,18)(15,21)(19,23)(20,24)(22,27)(26,28);(1,3,4,2)(6,9)(7,11)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27);(13,15,18,21,17,23,14,19)(16,22,24,27,25,28,20,26);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(6,9)(10,12)(13,14,17,18)(15,19,23,21)(16,24,25,20)(22,27,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240753, #72250956, #72257118, … |
| maximal normal subgroup of | #72240753, #72250956, #72257118, … |
| maximal quotient of | #72240764, #72343763 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296563);