#7897859

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

4708 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805055);