#8456833

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

5086 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,16,12)(17,18)(19,28)(20,22)(21,31)(23,27)(24,32)(25,30)(26,29);(3,4)(6,8)(9,11,13,16)(10,12,14,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,26)(27,28);(1,3,5,4)(2,6,7,8)(9,10)(11,12)(13,14)(15,16);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20,23,30)(18,25,27,22)(19,29,32,21)(24,26,28,31);(17,21,23,29)(18,26,27,31)(19,30,32,20)(22,24,25,28);(1,2)(3,6)(4,8)(5,7)(9,12,13,15)(10,11,14,16)(17,22,23,25)(18,20,27,30)(19,31,32,26)(21,24,29,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8364029);