#8372256

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	#35522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4501 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)(3,4,7,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,32,36)(27,30,31,33)(28,35,29,34)(37,40,39,38);(2,6)(3,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,32,31)(26,33,36,30)(28,37,29,39)(34,40,35,38);(1,3)(2,4)(5,7)(6,8)(12,20)(15,22)(18,24)(21,23)(25,28,32,29)(26,34,36,35)(27,37,31,39)(30,38,33,40);(1,4)(2,3)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29,32,28)(26,35,36,34)(27,37,31,39)(30,38,33,40);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,30,32,33)(26,27,36,31)(28,38,29,40)(34,37,35,39);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,31,32,27)(26,33,36,30)(28,37,29,39)(34,38,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,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, 8279452);