#8821826

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

4395 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)(3,4)(5,6)(7,8)(25,26,30,31)(27,32,35,29)(28,33,38,39)(34,40,36,37);(2,6)(4,7)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(26,31)(29,32)(33,39)(37,40);(2,6)(3,8)(10,12)(11,17)(16,20)(19,23)(25,27,30,35)(26,32,31,29)(28,36,38,34)(33,37,39,40);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28,30,38)(26,33,31,39)(27,34,35,36)(29,37,32,40);(1,4,5,7)(2,8,6,3)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29)(26,27)(28,37)(30,32)(31,35)(33,34)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=h,e^2=ghi,f^2=hi,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, 8729022);