#7927381

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

4429 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)(9,10,14,16)(11,15,17,12)(13,18,22,23)(19,21,24,20)(26,32)(28,34)(33,35)(36,38);(1,3,5,8)(2,6,4,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20)(25,26,31,32)(27,35,29,33)(28,30,34,40)(36,39,38,37);(1,2,5,4)(3,6,8,7)(10,16)(11,17)(18,23)(19,24)(25,27,31,29)(26,33,32,35)(28,36,34,38)(30,37,40,39);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,28,31,34)(26,30,32,40)(27,36,29,38)(33,37,35,39);(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24)(25,29,31,27)(26,33,32,35)(28,38,34,36)(30,37,40,39);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,18,16,23)(11,19,17,24)(12,20,15,21)(25,30)(26,34)(27,37)(28,32)(29,39)(31,40)(33,38)(35,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7834577);