#8285272

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	#44063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4538 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)(4,6)(5,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26)(27,38)(28,35)(29,34)(30,40)(31,36)(32,37)(33,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27)(26,33)(28,36)(29,30)(31,34)(32,39)(35,40)(37,38);(1,2)(3,5)(4,6)(7,8)(13,21)(16,23)(18,24)(20,22)(25,28,32,29)(26,34,37,35)(27,36,39,30)(31,38,40,33);(1,3,4,8)(2,5,6,7)(9,12,14,15)(10,11,17,19)(13,20,21,22)(16,18,23,24)(25,29,32,28)(26,35,37,34)(27,36,39,30)(31,38,40,33);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,30,32,36)(26,31,37,40)(27,29,39,28)(33,34,38,35);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31)(26,36)(27,34)(28,33)(29,38)(30,37)(32,40)(35,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192468);