#8760659

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	#53179	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4476 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8667855);