#8417422

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gi,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, 8324618);