#8898091

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

4704 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805287);