#8799069

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

4598 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gi,e^2=i,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, 8706265);