#8334295

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=gi,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, 8241491);