#8073122

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	#43108	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4300 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)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,26,31,34)(27,35,37,30)(28,32,36,40)(29,33,39,38);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24);(11,13)(16,18)(19,21)(23,24)(25,27,31,37)(26,30,34,35)(28,29,36,39)(32,38,40,33);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28)(26,32)(27,29)(30,38)(31,36)(33,35)(34,40)(37,39);(1,3,5,4)(2,6,8,7)(9,13,15,11)(10,18,14,16)(12,19,20,21)(17,23,22,24)(25,29)(26,33)(27,36)(28,37)(30,32)(31,39)(34,38)(35,40);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,18,13,16)(12,17,20,22)(19,23,21,24)(25,30)(26,27)(28,38)(29,32)(31,35)(33,36)(34,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7980318);