#8188121

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=ghi,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, 8095317);