#8417190

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

4413 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gi,e^2=gi,f^2=g,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, 8324386);