#8924659

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

4682 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,19,20,16)(12,17,18,13)(21,24,23,22)(25,26)(27,36)(28,32)(29,34)(30,35)(31,39)(33,38)(37,40);(1,2,4,6)(3,5,8,7)(10,15)(12,18)(16,19)(21,23)(25,27)(26,31)(28,37)(29,38)(30,39)(32,34)(33,40)(35,36);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(13,17)(21,23)(25,28,30,38)(26,32,35,33)(27,37,39,29)(31,34,36,40);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,16,15,19)(12,23,18,21)(13,22,17,24)(25,28)(26,33)(27,37)(29,39)(30,38)(31,40)(32,35)(34,36);(9,12,14,18)(10,17,15,13)(11,21,20,23)(16,22,19,24)(25,29,30,37)(26,34,35,40)(27,38,39,28)(31,32,36,33);(9,13,14,17)(10,18,15,12)(11,22,20,24)(16,21,19,23)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831855);