#8912614

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

4714 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=i,f^2=g,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, 8819810);