#8276452

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

4684 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,f^2=g,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, 8183648);