#8073143

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

4692 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)(10,15)(12,16)(18,23)(20,24)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,39)(37,40);(1,2,4,5)(3,6,7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(25,27)(26,32)(28,29)(30,36)(31,35)(33,40)(34,38)(37,39);(10,15)(12,16)(18,23)(20,24)(25,28,31,39)(26,33,36,38)(27,37,35,29)(30,40,32,34);(9,11)(10,16)(12,15)(13,21)(14,17)(18,20)(19,22)(23,24)(25,29)(26,34)(27,28)(30,33)(31,37)(32,38)(35,39)(36,40);(1,3,4,7)(2,6,5,8)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,28)(26,33)(27,29)(30,34)(31,39)(32,40)(35,37)(36,38);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,30)(26,35)(27,36)(28,34)(29,38)(31,32)(33,37)(39,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,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, 7980339);