#8087813

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

4378 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=g,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, 7995009);