#8982140

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,16,19,14)(12,13,17,23)(20,21,24,22)(26,32)(27,37)(28,30)(35,40);(3,8)(6,7)(9,11,15,19)(10,16,18,14)(12,20,17,24)(13,21,23,22)(25,26)(27,38)(28,34)(29,40)(30,36)(31,32)(33,37)(35,39);(1,3,5,8)(2,6,4,7)(11,19)(12,17)(13,23)(14,16)(25,27)(26,33)(28,29)(30,39)(31,37)(32,38)(34,35)(36,40);(3,8)(6,7)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,28,31,30)(26,34,32,36)(27,29,37,39)(33,35,38,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,21)(12,18)(14,20)(15,23)(16,24)(19,22)(25,29,31,39)(26,35,32,40)(27,28,37,30)(33,34,38,36);(1,4,5,2)(3,6,8,7)(9,14)(10,11)(12,22)(13,20)(15,16)(17,21)(18,19)(23,24)(25,30,31,28)(26,36,32,34)(27,39,37,29)(33,40,38,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8889336);