#8902764

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

5278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^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, 8809960);