#8088074

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

4415 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7995270);