#8095868

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
prime radical	#45476	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4421 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003064);