Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14734 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,9,7,20)(3,12,14,23,15,24,19,8)(4,16,17,10,18,26,21,25)(13,28,29,32,31,22,30,27);(1,3,7,14,6,15,5,19)(2,8,20,12,9,23,11,24)(4,13,21,29,18,31,17,30)(10,22,16,27,25,28,26,32);(2,9)(10,25)(11,20)(16,26);(1,4)(2,10)(3,13)(5,17)(6,18)(7,21)(8,22)(9,25)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(23,28)(24,32)Polycyclic presentation:
2^9|a^2=e,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215936, #72216154, #72216238, … |
| maximal normal subgroup of | #72215936, #72216154, #72216238, … |
| maximal quotient of | C2×#396548, #72256090, #72256130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303744);