#8231426

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	#45033	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8138622);