#7968860

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5231 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gh,e^2=g,f^2=gh,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, 7876056);