#8807308

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4279 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714504);