#7745322

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

5514 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi

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, 7652518);