#8434460

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
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8341656);