#8821846

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

4436 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)(3,4)(5,8)(6,7)(25,26,30,28)(27,32,34,31)(29,36,37,33)(35,38,40,39);(1,2,5,8)(3,6,7,4)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(27,34)(29,37)(31,32)(33,36);(1,2,5,8)(3,4,7,6)(10,12)(11,17)(16,20)(19,23)(25,27,30,34)(26,31,28,32)(29,35,37,40)(33,38,36,39);(1,3,5,7)(2,6,8,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,26)(27,31,34,32)(29,36,37,33)(35,38,40,39);(1,2)(3,6)(4,7)(5,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(1,4,5,6)(2,7,8,3)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,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, 8729042);