#7972950

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

4919 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)(3,8)(4,7)(5,6)(10,14)(15,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,30)(31,32);(2,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,27)(22,26)(23,24)(25,31)(28,30)(29,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,15,14,16)(18,23)(19,26)(20,29)(30,31);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,30)(28,31)(29,32);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,27)(21,31)(22,29)(23,28)(24,30)(26,32);(1,2)(3,7)(4,8)(5,6)(17,21,22,24)(18,19,23,26)(20,30,29,31)(25,32,28,27)

Polycyclic presentation:

2^9|c^2=ghi,d^2=g,e^2=g,f^2=h,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, 7880146);