Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14722 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,17)(9,19)(12,25)(14,23)(16,22)(20,28)(21,32)(24,27)(26,29)(30,31);(1,3,5,13,6,14,19,28)(2,7,9,11,10,20,18,23)(4,12,16,26,17,27,31,32)(8,21,22,24,15,29,30,25);(7,20)(11,23)(21,29)(24,25);(1,4,5,16,6,17,19,31)(2,8,9,22,10,15,18,30)(3,12,13,26,14,27,28,32)(7,21,11,24,20,29,23,25)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216166, #72216238, … |
| maximal normal subgroup of | #72215995, #72216166, #72216238, … |
| maximal quotient of | C2×#395172, #72252149, #72265264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302368);