#7972734

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,40,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,31)(28,37)(29,39)(30,36)(32,40)(33,38)(34,35);(1,3,5,4)(2,6,8,7)(10,16)(12,15)(17,23)(20,22)(27,36)(28,38)(31,35)(32,39);(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,40)(38,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,28,30,38)(26,32,34,39)(27,37,36,33)(29,31,40,35);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,35,38,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7879930);