#8167425

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

4450 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8074621);