Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2686 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15383 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1075 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)(3,11,13,5)(4,15,18,8)(6,10)(7,9)(12,29)(14,23,21,20)(16,32)(17,30)(19,28,22,26)(24,27)(25,31);(1,3)(2,5)(4,16)(6,21)(7,13)(8,25)(9,11)(10,23)(12,19)(14,24)(15,32)(17,26)(18,31)(20,27)(22,30)(28,29);(1,4)(2,8)(3,12)(5,17)(6,22)(7,18)(9,15)(10,28)(11,29)(13,30)(14,31)(16,21)(19,24)(20,32)(23,25)(26,27);(1,5)(2,3,9,13)(4,17)(6,23)(7,11)(8,12,15,30)(10,21,27,14)(16,26,31,28)(18,29)(19,32)(20,24)(22,25)Polycyclic presentation:
2^9|a^2=ef,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285398, #72286050, #72286126, … |
| maximal normal subgroup of | #72285398, #72286050, #72286126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335597);