#8359164

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8266360);