#8125511

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

4865 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)(14,16)(17,18,22,26)(19,27,29,23)(20,30,31,24)(21,25,28,32);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19,22,29)(18,23,26,27)(20,28,31,21)(24,32,30,25);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(20,31)(21,28)(24,30)(25,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,28)(21,29)(22,31)(23,32)(25,27)(26,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,28)(18,25,26,32)(19,20,29,31)(23,24,27,30);(1,3)(2,6)(4,7)(5,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^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, 8032707);