#8463745

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=hi,f^2=hi,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, 8370941);