#8109169

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 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)(3,7)(4,5)(6,8)(12,19)(16,21)(17,23)(22,24)(26,31)(28,35)(29,37)(34,39);(2,4)(6,7)(10,14)(15,20)(16,21)(22,24)(26,31)(27,33)(28,35)(36,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26,30,31)(27,29,33,37)(28,32,35,38)(34,36,39,40);(1,2,5,4)(3,6,8,7)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,27,30,33)(26,29,31,37)(28,34,35,39)(32,36,38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29)(26,33)(27,31)(28,36)(30,37)(32,39)(34,38)(35,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,e^2=g,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, 8016365);