#8322126

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	#59693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,f^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, 8229322);