#8716876

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

4823 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=gh,f^2=g,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, 8624072);