#8322347

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

4343 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8229543);