#8934863

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=gi,f^2=h,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, 8842059);