#8073166

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

4442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^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, 7980362);