#8175740

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

4602 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 8082936);