#8271967

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

4694 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)(4,8)(6,7)(10,11)(15,17)(16,18)(22,23)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,7)(3,4)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(27,28)(29,38)(31,34)(32,37);(2,7)(5,8)(13,21)(16,18)(19,24)(22,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,6,4)(2,8,7,5)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,3)(2,5,7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8179163);