#8390892

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

4640 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298088);