#10196060

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10103256);