#8127432

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

4996 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=g,e^2=h,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, 8034628);