#8712768

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	#59021	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8619964);