#8807306

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

4202 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 8714502);