#8119278

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

5070 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026474);