#8325495

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

4844 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 8232691);