#8402482

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	#35521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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)(3,7)(4,5)(6,8)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20)(26,28)(31,35)(32,36)(38,39);(2,5)(6,7)(9,10)(11,17)(12,19)(13,15)(14,16)(18,23)(20,24)(21,22)(26,28)(31,35)(32,36)(38,39);(1,2)(3,6)(4,5)(7,8)(13,21)(15,22)(18,24)(20,23)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,40,39);(9,13,14,21)(10,15,16,22)(11,18,19,24)(12,20,17,23)(25,29,30,37)(26,32,28,36)(27,33,34,40)(31,38,35,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8309678);