#8689876

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

5011 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)(17,18,21,24)(19,25,27,22)(20,23,30,29)(26,32,28,31);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,32)(29,30);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,32)(29,30);(2,7)(3,5)(10,14)(13,15)(17,19,21,27)(18,22,24,25)(20,28,30,26)(23,32,29,31);(1,4,6,8)(2,3,7,5)(9,11)(10,15)(12,16)(13,14)(17,18)(19,22)(20,29)(21,24)(23,30)(25,27)(26,32)(28,31);(1,5,6,3)(2,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,30)(18,23,24,29)(19,26,27,28)(22,31,25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=gh,f^2=g,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, 8597072);