#7746074

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4414 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653270);