#8371376

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	#45010	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8278572);