#10422971

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

4492 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)(11,12)(13,17)(14,18)(15,16)(19,20)(21,23)(22,24)(25,26,32,36)(27,37,40,33)(28,29,30,34)(31,38,39,35);(9,10,14,18)(11,15,16,12)(13,17,22,24)(19,23,21,20)(25,27,32,40)(26,33,36,37)(28,35,30,38)(29,39,34,31);(1,2)(3,5)(4,6)(7,8)(25,28,32,30)(26,34,36,29)(27,38,40,35)(31,33,39,37);(3,7)(5,8)(9,11)(10,15)(12,18)(13,21)(14,16)(17,20)(19,22)(23,24)(25,29)(26,28)(27,39)(30,36)(31,40)(32,34)(33,35)(37,38);(1,3)(2,5)(4,7)(6,8)(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,21,24,19)(25,30,32,28)(26,34,36,29)(27,38,40,35)(31,37,39,33);(1,2)(3,5)(4,6)(7,8)(9,13,14,22)(10,17,18,24)(11,19,16,21)(12,20,15,23)(25,31,32,39)(26,35,36,38)(27,29,40,34)(28,33,30,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10330167);