#8075288

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	#59018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,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, 7982484);