#7983179

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7890375);