#8109978

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

4756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017174);