#8325969

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,e^2=gh,f^2=hi,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, 8233165);