#7934510

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

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=ghi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7841706);