#8912630

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

4882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8819826);