#8161253

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

4818 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)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,31)(28,32);(1,3)(2,5)(4,8)(6,7)(10,11)(14,15)(18,23)(19,25)(26,29)(27,32);(1,2)(3,5)(4,6)(7,8)(9,10,13,11)(12,14,16,15)(18,23)(19,25)(26,29)(27,32);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,3)(2,5)(4,8)(6,7)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8068449);