#8254720

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 8161916);