Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(18,22)(19,20)(23,24);(3,8)(6,7)(11,16)(13,15)(17,19,21,20)(18,23,24,22);(10,14)(13,15)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72257332, #72257336, #72257516, … |
| maximal normal subgroup of | #72257332, #72257336, #72257516, … |
| maximal quotient of | #72308703, #72308752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6902970);