#8594956

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	#45563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4550 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,e^2=g,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, 8502152);