#8678863

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×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8586059);