#8471161

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

4611 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)(3,7)(4,6)(5,8)(9,10,15,12)(11,16,22,19)(13,23,21,17)(14,18,20,24)(26,32)(27,36)(28,38)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,18)(23,24)(25,26)(27,33)(28,30)(29,35)(31,32)(34,38)(36,40)(37,39);(9,10,15,12)(11,19,22,16)(13,17,21,23)(14,18,20,24)(25,27)(26,33)(28,37)(29,38)(30,39)(31,36)(32,40)(34,35);(1,2,4,6)(3,5,8,7)(9,12,15,10)(11,19,22,16)(13,23,21,17)(14,18,20,24)(25,28,31,38)(26,34,32,30)(27,37,36,29)(33,35,40,39);(1,3)(2,5)(4,8)(6,7)(9,13,15,21)(10,17,12,23)(11,20,22,14)(16,18,19,24)(25,29,31,37)(26,35,32,39)(27,38,36,28)(30,33,34,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,20)(10,18,12,24)(11,21,22,13)(16,17,19,23)(25,30,31,34)(26,28,32,38)(27,39,36,35)(29,33,37,40)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=hi,f^2=hi,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, 8378357);