#8508922

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	#57331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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)(3,7)(9,10,15,16)(11,17,12,20)(13,23,24,14)(18,22,21,19)(27,33)(30,36)(34,39)(37,40);(1,2,4,5)(3,6,7,8)(10,16)(13,24)(17,20)(18,21)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,20)(13,18,24,21)(14,19,23,22)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,17,16,20)(13,21,24,18)(14,19,23,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,3,4,7)(2,6,5,8)(9,13)(10,14)(11,18)(12,21)(15,24)(16,23)(17,19)(20,22)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,14)(10,13)(11,19)(12,22)(15,23)(16,24)(17,18)(20,21)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416118);