#7986354

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

4987 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)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18)(19,22)(20,28)(21,23)(24,29)(25,27)(26,32)(30,31);(2,5)(6,7)(9,10,13,14)(11,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(18,21)(19,25)(20,29)(30,32);(1,3)(2,6)(4,8)(5,7)(17,20,23,29)(18,24,21,28)(19,26,25,31)(22,30,27,32);(1,3)(2,7)(4,8)(5,6)(17,21,23,18)(19,27,25,22)(20,28,29,24)(26,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,22,23,27)(18,19,21,25)(20,30,29,32)(24,26,28,31)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=h,e^2=h,f^2=gi,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, 7893550);