#8105983

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5544 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013179);