#8461585

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

4546 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)(3,7)(4,6)(5,8)(12,20)(17,21)(18,24)(22,23)(25,26,32,35)(27,36,31,33)(28,30,29,34)(37,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,10,14,13)(11,16,19,15)(12,17,20,21)(18,23,24,22)(25,27,32,31)(26,33,35,36)(28,38,29,37)(30,39,34,40);(9,11)(10,15)(12,18)(13,16)(14,19)(17,22)(20,24)(21,23)(25,28,32,29)(26,34,35,30)(27,37,31,38)(33,39,36,40);(2,6)(5,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,23)(20,24)(21,22)(25,29,32,28)(26,30,35,34)(27,38,31,37)(33,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,17,13,21)(11,18,19,24)(15,22,16,23)(25,30)(26,28)(27,39)(29,35)(31,40)(32,34)(33,38)(36,37);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,16,19,15)(12,21,20,17)(18,23,24,22)(25,31,32,27)(26,33,35,36)(28,38,29,37)(30,40,34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=i,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, 8368781);