#8273480

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180676);