#8087812

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	#50278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 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)(11,18)(12,17)(13,19)(14,16)(15,22)(20,21)(23,24)(25,26)(27,29)(28,33)(30,34)(31,32)(35,37)(36,40)(38,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,18)(17,24)(19,23)(26,32)(27,28)(34,39)(35,36);(1,2,5,4)(3,6,8,7)(10,16)(13,17)(15,18)(21,24)(25,27,31,28)(26,29,32,33)(30,35,38,36)(34,37,39,40);(1,2,5,4)(3,6,8,7)(9,12,14,19)(10,17,16,13)(11,20,22,23)(15,24,18,21)(25,28,31,27)(26,33,32,29)(30,35,38,36)(34,37,39,40);(1,4,5,2)(3,7,8,6)(9,11)(10,18)(12,23)(13,21)(14,22)(15,16)(17,24)(19,20)(25,29,31,33)(26,28,32,27)(30,37,38,40)(34,36,39,35);(9,13,14,17)(10,19,16,12)(11,21,22,24)(15,23,18,20)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=g,e^2=gh,f^2=h,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, 7995008);