#7643368

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

4753 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550564);