#8081027

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	#37808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=g,f^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, 7988223);