Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14716 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2626 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)(3,13)(4,18)(5,12)(6,10)(7,11)(8,15)(9,19)(14,28)(16,24)(17,22)(20,26)(21,25)(23,30)(27,29)(31,32);(1,3,5,15,7,16,6,17)(2,8,10,13,11,22,12,24)(4,14,19,29,20,30,21,31)(9,23,18,27,25,28,26,32);(8,22)(13,24)(23,28)(27,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,23)(10,18)(11,25)(12,26)(13,27)(15,29)(16,30)(17,31)(22,28)(24,32)Polycyclic presentation:
2^9|b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72215958, #72216059, … |
| maximal normal subgroup of | #72215912, #72215958, #72216059, … |
| maximal quotient of | #72237518, C2×#394295, #72255951, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301491);