#8075432

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

4508 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,e^2=gh,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, 7982628);