#8159094

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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)(4,8)(5,6)(9,10,15,13)(11,18,17,14)(12,20,21,16)(19,22,23,24)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,24)(21,23)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,4)(5,8)(6,7)(12,21)(16,20)(19,23)(22,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(2,6)(3,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,23)(18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,10)(11,18,17,14)(12,16,21,20)(19,22,23,24)(25,27,31,30)(26,34,28,32)(29,35,38,37)(33,40,36,39);(1,4)(2,3)(5,7)(6,8)(9,14,15,18)(10,17,13,11)(12,22,21,24)(16,23,20,19)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8066290);