#8425144

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

4563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332340);