#8424965

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332161);