#8429576

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

5490 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336772);