#8188177

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095373);