#8206131

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	#50647	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=g,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, 8113327);