#8284518

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	#37752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4553 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8191714);