#8418141

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

4401 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:

(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26,32,31)(27,35,34,30)(28,38,29,33)(36,39,37,40);(2,3)(6,7)(9,11)(10,16)(12,14)(13,23)(15,20)(17,21)(18,22)(19,24)(25,27,32,34)(26,30,31,35)(28,36,29,37)(33,39,38,40);(1,3,5,2)(4,7,8,6)(11,20)(14,21)(16,24)(18,23)(25,28,32,29)(26,33,31,38)(27,36,34,37)(30,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,13)(11,21)(14,20)(15,17)(16,18)(19,22)(23,24)(25,29,32,28)(26,33,31,38)(27,37,34,36)(30,39,35,40);(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,23)(21,24)(25,30,32,35)(26,34,31,27)(28,39,29,40)(33,37,38,36);(9,14,15,21)(10,18,19,23)(11,12,20,17)(13,16,22,24)(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=h,e^2=h,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, 8325337);