#8934475

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841671);