#8924763

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
prime radical	#52012	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=ghi,e^2=ghi,f^2=h,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, 8831959);