#7973846

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

4937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=gh,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, 7881042);