#7899493

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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)(6,8)(9,10)(11,20)(12,22)(13,24)(14,19)(15,16)(17,23)(18,21)(25,26,31,34)(27,35,39,32)(28,29,38,40)(30,33,36,37);(1,2,4,5)(3,6,7,8)(9,11,15,19)(10,14,16,20)(12,21,23,13)(17,18,22,24)(25,27,31,39)(26,32,34,35)(28,30,38,36)(29,37,40,33);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,23)(18,24)(25,28)(26,29)(27,36)(30,39)(31,38)(32,33)(34,40)(35,37);(9,12)(10,17)(11,21)(13,19)(14,18)(15,23)(16,22)(20,24)(25,29)(26,28)(27,37)(30,32)(31,40)(33,39)(34,38)(35,36);(9,13,15,21)(10,18,16,24)(11,12,19,23)(14,22,20,17)(25,28)(26,29)(27,30)(31,38)(32,37)(33,35)(34,40)(36,39);(9,14)(10,19)(11,16)(12,18)(13,17)(15,20)(21,22)(23,24)(25,30)(26,33)(27,38)(28,39)(29,32)(31,36)(34,37)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7806689);