#7709457

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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616653);