#8188118

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

4390 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)(3,7)(4,5)(6,8)(10,15)(16,17)(18,22)(23,24)(25,26)(27,38)(28,31)(29,36)(30,35)(32,37)(33,40)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,12,17)(13,18,21,22)(19,23,20,24)(25,27)(26,33)(28,29)(30,39)(31,35)(32,40)(34,36)(37,38);(2,5)(6,7)(9,11,14,12)(10,16,15,17)(13,20,21,19)(18,24,22,23)(25,28,32,39)(26,34,37,31)(27,29,40,30)(33,36,38,35);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,29,32,30)(26,35,37,36)(27,28,40,39)(31,38,34,33);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,17,15,16)(13,20,21,19)(18,24,22,23)(25,30,32,29)(26,36,37,35)(27,39,40,28)(31,33,34,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,18,15,22)(11,19,12,20)(16,23,17,24)(25,31,32,34)(26,28,37,39)(27,35,40,36)(29,38,30,33)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095314);