Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20);(6,9)(10,12)(13,15)(14,17)(16,20)(18,19);(1,2,3,4)(15,20)(17,18);(13,15)(14,18)(16,20)(17,19);(15,20)(17,18);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|c^2=h,f^2=gi,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72233523, #72233538, #72233593, … |
| maximal normal subgroup of | #72233523, #72233538, #72233593, … |
| maximal quotient of | #72261393, #72262157, #72264709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537567);