#8467183

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

4457 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=h,e^2=hi,f^2=gi,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, 8374379);