#8167396

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

4476 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074592);