#8374196

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	#55347	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4289 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)(3,4)(5,6)(7,8)(9,10,15,19)(11,16,20,24)(12,13,17,23)(14,18,22,21)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(2,6)(3,7)(9,11)(10,16)(12,21)(13,14)(15,20)(17,18)(19,24)(22,23)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(1,3,5,7)(2,4,6,8)(11,20)(14,22)(16,24)(18,21)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,13)(11,21)(14,16)(15,17)(18,20)(19,23)(22,24)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(9,13)(10,17)(11,22)(12,19)(14,20)(15,23)(16,21)(18,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(9,14,15,22)(10,18,19,21)(11,13,20,23)(12,16,17,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281392);