#8188154

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(11,18)(12,15)(13,20)(14,17)(16,22)(19,21)(23,24)(25,26)(27,34)(28,37)(29,38)(30,32)(31,33)(35,36)(39,40);(1,2)(3,5)(4,6)(7,8)(10,12)(11,17)(16,20)(19,23)(25,27,31,30)(26,32,33,34)(28,35,38,40)(29,36,37,39);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,29)(27,35)(30,40)(31,38)(32,36)(33,37)(34,39);(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,29)(26,28)(27,36)(30,39)(31,37)(32,35)(33,38)(34,40);(1,3)(2,5)(4,8)(6,7)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,28)(27,36)(30,39)(31,37)(32,35)(33,38)(34,40);(1,2)(3,5)(4,6)(7,8)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,27)(26,32,33,34)(28,35,38,40)(29,39,37,36)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=ghi,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, 8095350);