#7968222

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4665 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7875418);