#8127637

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	#59173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034833);