#8324779

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	#39410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,e^2=g,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, 8231975);