#8110601

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,f^2=ghi,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, 8017797);