#9057920

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,6)(3,7,8,4)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,30,34)(27,31,29,33)(28,32,38,37)(35,39,36,40);(2,6)(4,7)(10,15)(12,16)(18,22)(20,23)(25,27,30,29)(26,31,34,33)(28,36,38,35)(32,40,37,39);(1,2)(3,4)(5,6)(7,8)(11,17)(12,16)(13,21)(18,22)(25,26,30,34)(27,33,29,31)(28,37,38,32)(35,39,36,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,26)(27,31)(28,37)(29,33)(30,34)(32,38)(35,40)(36,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,40)(34,37);(1,3)(2,4)(5,8)(6,7)(9,13,14,21)(10,18,15,22)(11,19,17,24)(12,20,16,23)(25,29,30,27)(26,33,34,31)(28,36,38,35)(32,40,37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965116);