#8994187

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	#37801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gi,e^2=gh,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, 8901383);