#8087990

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

4255 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)(9,10,15,17)(11,19,12,16)(13,14,18,24)(20,23,22,21)(25,26,31,35)(27,36,28,32)(29,30,34,33)(37,40,39,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,17,19)(13,20,18,22)(14,21,24,23)(27,28)(32,36)(37,39)(38,40);(1,3,5,4)(2,6,8,7)(10,17)(14,24)(16,19)(21,23)(25,27,31,28)(26,32,35,36)(29,39,34,37)(30,40,33,38);(9,12,15,11)(10,16,17,19)(13,20,18,22)(14,23,24,21)(25,28,31,27)(26,32,35,36)(29,39,34,37)(30,38,33,40);(9,13,15,18)(10,14,17,24)(11,20,12,22)(16,21,19,23)(25,29,31,34)(26,33,35,30)(27,37,28,39)(32,40,36,38);(1,4,5,3)(2,7,8,6)(9,14)(10,18)(11,21)(12,23)(13,17)(15,24)(16,22)(19,20)(25,30,31,33)(26,34,35,29)(27,38,28,40)(32,39,36,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 7995186);