#8701597

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

4911 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608793);