#8599513

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

4588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506709);