#8421863

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

5126 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)(3,5)(4,8)(6,7)(18,19)(23,25)(24,26)(30,31);(1,3,6,8)(2,4,7,5)(9,10)(11,14)(12,15)(13,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31);(1,4,6,5)(2,3,7,8)(9,11)(10,13)(12,16)(14,15);(2,7)(3,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31);(1,5,6,4)(2,8,7,3)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,2)(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gh,e^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, 8329059);