#8108740

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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)(5,7)(9,10,15,18)(11,16,13,12)(14,17,22,24)(19,23,21,20)(25,26,31,34)(27,29,37,32)(28,30,38,33)(35,40,39,36);(2,4)(3,8)(9,11,15,13)(10,12,18,16)(14,21,22,19)(17,23,24,20)(25,27,31,37)(26,32,34,29)(28,35,38,39)(30,36,33,40);(1,2,6,4)(3,7,8,5)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,28,31,38)(26,33,34,30)(27,35,37,39)(29,36,32,40);(1,3,6,8)(2,7,4,5)(9,13,15,11)(10,16,18,12)(14,21,22,19)(17,23,24,20)(25,27)(26,32)(28,35)(29,34)(30,36)(31,37)(33,40)(38,39);(1,4,6,2)(3,5,8,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,31,32)(26,27,34,37)(28,36,38,40)(30,39,33,35);(1,5)(2,3)(4,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,30)(26,28)(27,36)(29,39)(31,33)(32,35)(34,38)(37,40)

Polycyclic presentation:

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

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, 8015936);