Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,17)(5,6)(8,13)(9,23)(10,19)(14,29)(15,16)(21,30)(22,26)(27,31)(28,32);(1,2)(3,13)(4,9)(5,10)(6,19)(7,11)(8,16)(12,22)(14,30)(15,26)(17,23)(18,24)(20,25)(21,27)(28,31)(29,32);(1,3)(2,8)(4,14)(5,16)(6,12)(7,15)(9,21)(10,26)(11,22)(13,19)(17,31)(18,29)(20,27)(23,32)(24,30)(25,28);(1,4,5,18,7,17,6,20)(2,9,10,24,11,23,19,25)(3,14,16,29,15,31,12,27)(8,21,26,30,22,32,13,28)Polycyclic presentation:
2^9|d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318883, #72318935, #72318991, … |
| maximal normal subgroup of | #72318883, #72318935, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335027);