#8416824

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

4403 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)(4,5)(6,8)(11,17)(15,18)(19,21)(23,24)(25,26,32,31)(27,35,34,30)(28,38,29,33)(36,39,37,40);(2,5)(6,7)(9,10)(11,18)(12,20)(13,14)(15,17)(16,22)(19,23)(21,24)(25,27,32,34)(26,30,31,35)(28,36,29,37)(33,39,38,40);(1,2)(3,6)(4,5)(7,8)(10,14)(15,18)(16,20)(23,24)(25,28,32,29)(26,33,31,38)(27,36,34,37)(30,39,35,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,29,32,28)(26,33,31,38)(27,37,34,36)(30,39,35,40);(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,30,32,35)(26,34,31,27)(28,39,29,40)(33,37,38,36);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,31,32,26)(27,35,34,30)(28,33,29,38)(36,39,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324020);