#8745956

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653152);