#8291621

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	#37448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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)(11,16)(12,14)(13,17)(15,19)(18,23)(20,21)(22,24)(25,26,32,36)(27,37,38,33)(28,35,29,34)(30,40,39,31);(2,6)(3,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27,32,38)(26,33,36,37)(28,30,29,39)(31,34,40,35);(1,3)(2,4)(5,7)(6,8)(13,21)(17,20)(18,24)(22,23)(25,28,32,29)(26,34,36,35)(27,30,38,39)(31,33,40,37);(1,4)(2,3)(5,8)(6,7)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,29,32,28)(26,35,36,34)(27,30,38,39)(31,33,40,37);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,30)(26,31)(27,29)(28,38)(32,39)(33,34)(35,37)(36,40);(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,31)(26,30)(27,34)(28,37)(29,33)(32,40)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198817);