#8716724

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

4939 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)(17,18,23,27)(19,22,26,24)(20,25,30,21)(28,29,32,31);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,12,15,14)(17,19,23,26)(18,24,27,22)(20,28,30,32)(21,29,25,31);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,16)(13,15)(17,20,23,30)(18,25,27,21)(19,28,26,32)(22,29,24,31);(2,6)(4,7)(9,12,13,14)(10,15,16,11)(17,20)(18,21)(19,28)(22,31)(23,30)(24,29)(25,27)(26,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,15)(14,16)(17,21)(18,20)(19,29)(22,32)(23,25)(24,28)(26,31)(27,30);(1,4,5,7)(2,8,6,3)(9,12)(10,15)(11,16)(13,14)(17,22,23,24)(18,26,27,19)(20,29,30,31)(21,28,25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=i,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, 8623920);