#8075543

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=h,f^2=h,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, 7982739);