#8618186

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	#48050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525382);