#8509612

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

5024 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8416808);