Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14715 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,9)(5,17)(6,11)(7,13)(10,20)(12,28)(14,24)(15,27)(16,26)(18,22)(19,25)(21,29)(23,30)(31,32);(1,3)(2,7)(4,12)(5,18)(6,14)(8,17)(9,21)(10,24)(11,22)(13,20)(15,31)(16,30)(19,29)(23,25)(26,32)(27,28);(1,3)(2,8)(4,12)(5,18)(6,14)(7,10)(9,23)(11,24)(13,20)(15,31)(16,30)(17,22)(19,29)(21,27)(25,32)(26,28);(1,4)(2,9)(3,12)(5,19)(6,16)(7,21)(8,23)(10,27)(11,26)(13,31)(14,30)(15,20)(17,25)(18,29)(22,32)(24,28)Polycyclic presentation:
2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219688, #72219692, … |
| maximal normal subgroup of | #72215953, #72219688, #72219692, … |
| maximal quotient of | #72260756, #72261029, #72261041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301194);