#8137322

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,19,22,16)(12,23,21,17)(14,24,20,18)(28,36)(32,39)(34,40)(37,38);(2,3)(6,7)(9,11)(10,16)(12,20)(13,19)(14,21)(15,22)(17,18)(23,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,2,5,3)(4,6,8,7)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,5,2)(4,7,8,6)(9,12,15,21)(10,17,13,23)(11,20,22,14)(16,18,19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13,15,10)(11,19,22,16)(12,17,21,23)(14,24,20,18)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40);(9,14,15,20)(10,18,13,24)(11,21,22,12)(16,17,19,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044518);