#9054408

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	#48741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961604);