#8459158

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

4802 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 8366354);