#8810235

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	#56132	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4348 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717431);