#8077083

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×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4429 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=i,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, 7984279);