#8446019

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

4374 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353215);