Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417688 |
| 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 | |
2405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(5,6)(9,23)(10,22)(11,12)(13,25)(14,28)(15,21)(17,26)(18,20)(24,27)(31,32);(1,2)(3,13)(4,17)(5,12)(6,8)(7,11)(9,15)(10,18)(14,24)(16,23)(19,22)(20,26)(21,25)(27,29)(28,32)(30,31);(1,3)(2,9)(4,14)(5,15)(6,21)(7,16)(8,13)(10,24)(11,25)(12,23)(17,31)(18,29)(19,28)(20,30)(22,27)(26,32);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,22)(9,24)(11,26)(12,17)(13,27)(15,30)(16,28)(21,29)(23,31)(25,32)Polycyclic presentation:
2^9|e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219488, #72219491, … |
| maximal normal subgroup of | #72213913, #72219488, #72219491, … |
| maximal quotient of | #72251172, #72255763 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324884);