Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10)(11,18)(12,15)(13,17)(14,22)(16,20)(19,21)(23,24);(1,2)(3,4)(6,7)(9,10)(11,19)(12,16)(13,17)(14,23)(15,20)(18,21)(22,24);(2,4)(5,6,8,7)(9,11,20,21,13,23,12,22)(10,14,15,24,17,19,16,18);(2,4)(5,7,8,6)(9,10)(11,14)(12,16)(13,17)(15,20)(18,22)(19,23)(21,24);(9,12,13,20)(10,15,17,16)(11,21,23,22)(14,18,19,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307591, #72307781, #72307900 |
| maximal normal subgroup of | #72307591, #72307781, #72307900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351481);