#8994103

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	#42063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8901299);