#8746596

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

4341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653792);