#8325441

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gh,e^2=gh,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, 8232637);