#8688354

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

4925 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595550);