#8295543

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	#40517	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4457 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)(3,4)(5,6)(7,8)(25,26,32,38)(27,36,30,33)(28,35,29,34)(31,37,40,39);(2,6)(3,7)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,27,32,30)(26,33,38,36)(28,39,29,37)(31,35,40,34);(1,3,5,7)(2,4,6,8)(10,15)(13,22)(16,23)(20,24)(25,28,32,29)(26,34,38,35)(27,39,30,37)(31,36,40,33);(1,4)(2,7)(3,6)(5,8)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,29,32,28)(26,35,38,34)(27,39,30,37)(31,36,40,33);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,30,32,27)(26,36,38,33)(28,37,29,39)(31,34,40,35);(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,31)(26,37)(27,35)(28,33)(29,36)(30,34)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,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, 8202739);