#8106062

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

4527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013258);