#9062551

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	#44842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969747);