#8091754

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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)(9,10,15,18)(11,19,21,14)(12,16,20,24)(13,17,23,22)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(1,2,5,4)(3,6,8,7)(9,11,15,21)(10,14,18,19)(12,13,20,23)(16,22,24,17)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(9,12)(10,16)(11,13)(14,22)(15,20)(17,19)(18,24)(21,23)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(9,13)(10,17)(11,20)(12,21)(14,16)(15,23)(18,22)(19,24)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,38,40);(1,4,5,2)(3,7,8,6)(9,14)(10,11)(12,22)(13,16)(15,19)(17,20)(18,21)(23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=i,f^2=gi,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, 7998950);