#8421845

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	#59142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4675 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329041);