#7985169

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

5606 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,28)(20,21)(22,27)(23,25)(24,30)(26,31)(29,32);(2,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,23,30)(18,24,25,28)(20,27,26,32)(21,29,31,22);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,11,16,15)(18,25)(20,26)(24,28)(27,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,23,26)(18,21,25,31)(19,27,30,32)(22,24,29,28);(1,3)(2,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,21)(18,26)(19,29)(20,25)(22,30)(23,31)(24,32)(27,28);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,15,16,11)(17,22,23,29)(18,27,25,32)(19,21,30,31)(20,24,26,28)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=h,f^2=gi,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, 7892365);