#8075252

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982448);