#7975050

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

4918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,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, 7882246);