#8226705

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4275 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:

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

Polycyclic presentation:

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

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, 8133901);