#10529226

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

4572 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10436422);