#10094941

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,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, 10002137);