#8125705

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

4905 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)(10,13)(12,16)(18,22)(19,25)(24,29)(26,31);(1,2)(3,6)(4,5)(7,8)(9,10,14,13)(11,12,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(2,5)(6,8)(9,11,14,15)(10,12,13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,13,14,10)(11,16,15,12)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=gh,f^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,e^c=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, 8032901);