#8429658

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	#40720	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4620 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)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(28,36)(32,39)(34,40)(37,38);(3,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8336854);