#8720725

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

5061 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,21,26,23)(20,24,30,25)(28,29,32,31);(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,28)(21,27)(22,26)(24,31)(25,29)(30,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,30)(18,24,27,25)(19,28,26,32)(21,29,23,31);(2,6)(4,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,31)(22,30)(23,29)(24,27)(26,32);(1,3)(2,7)(4,6)(5,8)(17,19)(18,21)(20,28)(22,26)(23,27)(24,29)(25,31)(30,32);(1,4,5,7)(2,8,6,3)(17,21)(18,26)(19,27)(20,29)(22,23)(24,32)(25,28)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=i,f^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, 8627921);