#8256621

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
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163817);