Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213×C4 |
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 | |
13866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8);(1,2,3,4)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24);(5,6,7,8)(10,14)(15,20)(16,21)(22,24);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(11,17)(15,20)(18,23)(22,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,f^2=i,b^a=bg,c^a=ch,c^b=ci,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238208, #72238219, #72238251, … |
| maximal normal subgroup of | #72238208, #72238219, #72238251, … |
| maximal quotient of | C2×#7629697, D4×#2397, #72280449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536893);