#7932381

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

4689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7839577);