#8195701

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

4285 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=h,f^2=gi,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, 8102897);