#8618165

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	#59060	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4498 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525361);