#7864193

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=gh,e^2=g,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, 7771389);