#8165253

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

5180 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)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,28)(29,32);(1,2,4,6)(3,5,7,8)(10,14)(11,15)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,30,32);(1,3)(2,5)(4,7)(6,8)(9,11,13,15)(10,12,14,16)(18,24)(20,25)(26,30)(28,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32);(1,3)(2,5)(4,7)(6,8)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28);(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=gi,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^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, 8072449);