Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,15,16,14,18,13,17)(11,19,20,12,21,24,22,23);(5,6)(7,8)(9,11)(10,12)(13,20)(14,21)(15,22)(16,19)(17,24)(18,23);(2,4)(6,8)(9,12)(10,11)(13,19)(14,23)(15,24)(16,20)(17,22)(18,21);(2,4)(10,16)(11,20)(13,15)(17,18)(19,24)(21,22);(11,21)(12,23)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72315629, #72315721, #72315877 |
| maximal normal subgroup of | #72315629, #72315721, #72315877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410552);