#8080989

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

4650 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7988185);