Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15310 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1928 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,29)(15,23)(17,28)(18,22)(19,26)(20,25)(21,27)(24,30)(31,32);(1,3,5,16,7,17,6,15)(2,8,11,13,12,23,10,28)(4,18,19,30,21,31,20,14)(9,24,25,22,27,29,26,32);(1,4)(2,9)(3,14,17,30)(5,20)(6,19)(7,21)(8,22,23,32)(10,25)(11,26)(12,27)(13,24,28,29)(15,18,16,31);(2,10)(3,15)(5,6)(8,23)(9,25)(11,12)(14,18)(16,17)(19,20)(22,32)(26,27)(30,31)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,g^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,f^b=fi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215995, #72216077, … |
| maximal normal subgroup of | #72213514, #72215995, #72216077, … |
| maximal quotient of | #72237174, #72239600, #72250591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330099);