#7966751

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

5472 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)(11,13)(12,15)(14,16)(17,18)(19,24)(20,29)(21,23)(22,26)(25,30)(27,28)(31,32);(2,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,27)(18,24,21,28)(20,22,30,32)(25,31,29,26);(1,2,6,5)(3,7,4,8)(9,10)(11,14)(12,15)(13,16)(18,21)(19,27)(20,30)(26,31);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,22)(21,29)(23,30)(24,31)(26,28)(27,32);(1,4,6,3)(2,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,28,27,24)(20,29,30,25)(22,26,32,31);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,15)(13,16)(17,22)(18,26)(19,20)(21,31)(23,32)(24,29)(25,28)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=g,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, 7873947);