#7987310

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	#59657	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4696 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)(3,6,4,7)(10,12)(14,15)(17,18)(19,22)(20,27)(21,29)(23,24)(25,30)(26,31)(28,32);(1,2)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,19,23,20)(18,22,24,27)(21,26,30,28)(25,32,29,31);(9,10,13,12)(11,15,16,14)(18,24)(19,20)(21,30)(31,32);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(17,20,23,19)(18,22,24,27)(21,28,30,26)(25,32,29,31);(1,3,5,4)(2,6,8,7)(9,12,13,10)(11,15,16,14)(17,21)(18,25)(19,26)(20,28)(22,32)(23,30)(24,29)(27,31);(1,4,5,3)(2,7,8,6)(17,22)(18,19)(20,24)(21,31)(23,27)(25,28)(26,29)(30,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gi,e^2=g,f^2=ghi,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, 7894506);