Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7503371 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2679 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,8,12)(3,15)(4,19)(5,24,25,11)(6,13,7,14)(9,17)(10,21)(16,27)(18,28)(20,32,26,29)(22,30)(23,31);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,29)(11,31)(12,27)(13,28)(14,15)(19,24)(22,25)(23,26)(30,32);(1,4)(2,10)(3,5)(6,21)(7,22)(8,23)(9,11)(12,30)(13,31)(14,19)(15,29)(16,20)(17,26)(18,25)(24,27)(28,32);(1,5)(2,11)(3,4)(6,26)(7,20)(8,25)(9,10)(12,24)(13,32)(14,29)(15,19)(16,22)(17,21)(18,23)(27,30)(28,31);(2,12)(9,27)(10,30)(11,24)(13,14)(15,28)(19,31)(29,32)Polycyclic presentation:
2^9|a^2=fg,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214082, #72219482, #72219488, … |
| maximal normal subgroup of | #72214082, #72219482, #72219488, … |
| maximal quotient of | #72258208, #72258209, #72258215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410567);