#8592179

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

5143 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8499375);