#8920742

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

4619 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,19,20,16)(12,17,18,13)(21,24,23,22)(25,26,31,33)(27,34,29,32)(28,37,38,30)(35,36,39,40);(2,5)(6,7)(10,15)(12,18)(16,19)(21,23)(25,27,31,29)(26,32,33,34)(28,35,38,39)(30,36,37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(13,17)(21,23)(28,38)(30,37)(35,39)(36,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,20)(10,16,15,19)(12,23,18,21)(13,22,17,24)(25,28)(26,30)(27,35)(29,39)(31,38)(32,36)(33,37)(34,40);(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,29,31,27)(26,32,33,34)(28,35,38,39)(30,40,37,36);(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,38)(34,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827938);