#8716890

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

5125 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8624086);