#8075471

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

4324 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)(9,10,15,14)(11,13,18,16)(12,17,21,23)(19,22,24,20)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11,15,18)(10,16,14,13)(12,19,21,24)(17,20,23,22)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,2,5,4)(3,6,8,7)(11,18)(13,16)(19,24)(20,22)(25,27,30,28)(26,32,34,31)(29,35,37,36)(33,39,40,38);(9,12)(10,17)(11,19)(13,22)(14,23)(15,21)(16,20)(18,24)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(9,13)(10,11)(12,20)(14,18)(15,16)(17,24)(19,23)(21,22)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(1,4,5,2)(3,7,8,6)(9,14,15,10)(11,13,18,16)(12,17,21,23)(19,20,24,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=h,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, 7982667);