#8109897

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4358 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)(3,8)(4,7)(5,6)(13,21)(17,22)(18,23)(20,24)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,6)(7,8)(10,15)(12,19)(17,22)(20,24)(26,30)(28,37)(31,33)(34,40);(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8017093);