#8552406

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	#37343	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4248 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8459602);