#8273479

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	#40696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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)(27,31)(28,35)(29,34)(30,39)(32,37)(33,38)(36,40);(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)(26,33,37,31)(28,30,29,40)(34,39,35,36);(2,5)(6,8)(10,15)(11,17)(16,21)(18,24)(25,28,32,29)(26,34,37,35)(27,30,38,40)(31,36,33,39);(3,7)(6,8)(9,12)(10,11)(13,22)(14,20)(15,17)(16,24)(18,21)(19,23)(25,29,32,28)(26,35,37,34)(27,30,38,40)(31,36,33,39);(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)(26,36)(27,29)(28,38)(31,35)(32,40)(33,34)(37,39);(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)(26,27)(28,39)(29,36)(30,35)(32,33)(34,40)(37,38)

Polycyclic presentation:

2^9|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, 8180675);