Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442597 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(7,20)(11,23)(21,25)(24,31);(1,2)(3,11)(4,15)(5,18)(6,9)(7,14)(8,17)(10,19)(12,25)(13,23)(16,29)(20,28)(21,26)(22,30)(24,27)(31,32);(1,3,5,14,6,13,19,28)(2,7,10,11,9,20,18,23)(4,12,17,27,16,26,30,32)(8,21,15,24,22,25,29,31);(1,4)(2,8,9,22)(3,12)(5,17)(6,16)(7,21,20,25)(10,15,18,29)(11,24,23,31)(13,26)(14,27)(19,30)(28,32);(4,16)(8,22)(12,26)(15,29)(17,30)(21,25)(24,31)(27,32)Polycyclic presentation:
2^9|c^2=h,d^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,g^b=gi,h^b=hi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216077, #72216238, … |
| maximal normal subgroup of | #72215721, #72216077, #72216238, … |
| maximal quotient of | #72250791, C2×#6461519, #72255984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368715);