#8372187

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

4647 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8279383);