#8188153

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

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

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, 8095349);