#7983183

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

4737 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)(3,8)(4,7)(5,6)(9,10,13,12)(11,15,16,14)(19,25)(23,29)(26,31)(30,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(2,6)(3,7)(9,12,13,10)(11,15,16,14)(18,22)(23,29)(24,27)(30,32);(1,2,5,6)(3,4,7,8)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,3)(2,4)(5,7)(6,8)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,4)(2,7)(3,6)(5,8)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,f^2=hi,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, 7890379);