#8075635

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

4495 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982831);