#8119296

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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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)(3,7)(4,6)(5,8)(9,10,14,16)(11,18,21,15)(12,22,17,13)(19,20,23,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,38)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,16,18)(12,19,17,23)(13,20,22,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(12,17)(15,18)(19,23)(25,27,30,28)(26,32,34,31)(29,35,38,36)(33,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,23)(22,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,12,14,17)(10,13,16,22)(11,19,21,23)(15,20,18,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(9,13)(10,17)(11,20)(12,16)(14,22)(15,23)(18,19)(21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026492);