#8772649

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

4697 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,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, 8679845);