#8075398

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

4409 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)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,3,5,8)(2,6,4,7)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,2,5,4)(3,6,8,7)(10,15)(13,22)(16,23)(19,24)(27,29)(32,34)(35,37)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,2,5,4)(3,6,8,7)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19)(25,29,31,27)(26,34,30,32)(28,35,36,37)(33,39,38,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,30,31,26)(27,34,29,32)(28,33,36,38)(35,39,37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,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, 7982594);