#8421666

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

5099 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328862);