#8912475

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

5064 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)(17,18,23,27)(19,22,29,24)(20,30,28,25)(21,31,32,26);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,29)(18,24,27,22)(20,21,28,32)(25,31,30,26);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,20,23,28)(18,25,27,30)(19,21,29,32)(22,26,24,31);(11,14)(12,16)(17,21,23,32)(18,26,27,31)(19,28,29,20)(22,30,24,25);(1,3)(2,5)(4,7)(6,8)(9,12,13,16)(10,14,15,11)(17,19)(18,22)(20,21)(23,29)(24,27)(25,26)(28,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,22)(18,19)(20,31)(21,30)(23,24)(25,32)(26,28)(27,29)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,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, 8819671);