#7866460

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	#45271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4420 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7773656);