Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417566 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(7,14)(8,9)(11,15)(12,18)(16,17)(19,20);(1,2)(3,4)(5,6)(7,15)(8,12)(9,18)(10,13)(11,17)(14,19)(16,20);(5,7)(6,11)(8,17)(9,14)(10,16)(12,20)(13,19)(15,18);(2,4)(8,9)(12,18)(14,17)(15,20)Polycyclic presentation:
2^9|f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210524, #72216141, #72216144, … |
| maximal normal subgroup of | #72210524, #72216141, #72216144, … |
| maximal quotient of | #72233854, #72261993, C2×#417566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324762);