#8455719

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

4351 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,e^2=ghi,f^2=h,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, 8362915);