#9453054

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	#47723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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)(3,6)(4,5)(7,8)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,20,23)(12,13,17,22)(18,24,21,19)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,7)(4,6)(5,8)(10,15)(13,22)(16,23)(19,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,18)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,17)(10,13,15,22)(11,18,20,21)(16,24,23,19);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,19)(12,15)(14,22)(16,18)(20,24)(21,23)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9360250);