#8288171

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

5030 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8195367);