#8374853

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=h,f^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,e^d=egi

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, 8282049);