#8898168

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

5080 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)(11,13)(12,16)(14,15)(17,18,23,25)(19,24,27,29)(20,22,30,31)(21,26,32,28);(1,2)(3,5)(4,6)(7,8)(10,15)(11,16)(17,19,23,27)(18,24,25,29)(20,28,30,26)(21,31,32,22);(1,2,4,6)(3,5,7,8)(9,11,14,16)(10,12,15,13)(18,25)(19,27)(22,31)(26,28);(3,7)(5,8)(9,12,14,13)(10,11,15,16)(17,20)(18,22)(19,28)(21,24)(23,30)(25,31)(26,27)(29,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,21)(18,26)(19,22)(20,29)(23,32)(24,30)(25,28)(27,31);(1,2)(3,5)(4,6)(7,8)(9,13,14,12)(10,16,15,11)(17,22)(18,20)(19,21)(23,31)(24,28)(25,30)(26,29)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,f^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, 8805364);