#8165211

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

5482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8072407);