#7852856

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7760052);