#8127512

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

5086 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)(19,24)(20,22)(21,30)(23,25)(26,32)(27,31)(28,29);(1,2,6,5)(3,7,8,4)(11,15)(14,16)(17,19,23,21)(18,24,25,30)(20,29,26,27)(22,28,32,31);(3,8)(4,7)(9,10,13,12)(11,14,15,16)(18,25)(19,21)(20,26)(28,31);(1,3,6,8)(2,7,5,4)(9,11,13,15)(10,14,12,16)(17,20,23,26)(18,22,25,32)(19,27,21,29)(24,31,30,28);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,16,15,14)(17,21,23,19)(18,24,25,30)(20,29,26,27)(22,31,32,28);(1,5,6,2)(3,4,8,7)(17,22)(18,26)(19,28)(20,25)(21,31)(23,32)(24,27)(29,30)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=gh,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, 8034708);