#8075448

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

4432 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=h,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, 7982644);