Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417628 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2409 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)(3,8)(4,9)(5,11)(6,10)(7,12)(13,29)(14,23)(15,22)(16,32)(17,27)(18,26)(19,24)(20,28)(21,30)(25,31);(1,3)(2,8)(4,16)(5,19)(6,15)(7,14)(9,25)(10,22)(11,24)(12,23)(13,18)(17,31)(20,30)(21,26)(27,32)(28,29);(1,4)(2,9)(3,13)(5,20)(6,18)(7,17)(8,21)(10,26)(11,28)(12,27)(14,30)(15,31)(16,19)(22,32)(23,29)(24,25);(2,10)(5,6)(8,22)(9,26)(11,12)(15,19)(16,31)(18,20)(21,32)(23,24)(25,29)(27,28)Polycyclic presentation:
2^9|g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216037, #72216123, … |
| maximal normal subgroup of | #72215958, #72216037, #72216123, … |
| maximal quotient of | #72239603, #72239625, C2×#417628 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324824);