#8188142

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	#40696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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)(10,11)(12,19)(16,18)(20,24)(25,26)(27,35)(28,39)(29,40)(30,34)(31,32)(33,36)(37,38);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(26,32)(27,36)(28,40)(34,38);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27,31,36)(26,33,32,35)(28,37,40,30)(29,38,39,34);(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28)(26,29)(27,37)(30,36)(31,40)(32,39)(33,38)(34,35);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,28)(27,38)(30,35)(31,39)(32,40)(33,37)(34,36);(1,2)(3,5)(4,6)(7,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,37)(26,34,32,38)(27,28,36,40)(29,35,39,33)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095338);