#8622203

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	#41172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4313 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529399);