#8291613

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4491 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)(3,8)(4,6)(5,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26,31,35)(27,32,38,39)(28,37,29,33)(30,34,40,36);(1,2,6,4)(3,8,5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27,31,38)(26,32,35,39)(28,34,29,36)(30,37,40,33);(1,2,6,4)(3,7,5,8)(11,17)(15,18)(19,23)(22,24)(25,28,31,29)(26,33,35,37)(27,36,38,34)(30,32,40,39);(1,3,6,5)(2,7,4,8)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,31,28)(26,33,35,37)(27,34,38,36)(30,32,40,39);(1,4,6,2)(3,7,5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,32)(28,33)(29,37)(30,34)(31,35)(36,40)(38,39);(1,5,6,3)(2,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30)(26,34)(27,37)(28,39)(29,32)(31,40)(33,38)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gi,e^2=g,f^2=g,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, 8198809);