#8285483

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	#52335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(3,4)(5,7)(6,8)(9,10,15,18)(11,16,19,14)(12,13,17,23)(20,21,24,22)(25,26,31,34)(27,29,33,30)(28,36,39,32)(35,38,40,37);(1,3)(2,6)(4,7)(5,8)(9,11,15,19)(10,16,18,14)(12,20,17,24)(13,21,23,22)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(2,7)(4,6)(11,19)(14,16)(20,24)(21,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(2,7)(3,8)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,23)(19,24)(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,21)(12,18)(14,20)(15,23)(16,24)(19,22)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,4,5,6)(2,3,7,8)(9,14)(10,11)(12,22)(13,20)(15,16)(17,21)(18,19)(23,24)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=gh,f^2=g,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, 8192679);