#8445968

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

4532 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8353164);