#8132500

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	#42894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8039696);