#9028050

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
prime radical	#47685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935246);