#8197588

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

4923 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)(17,18,23,24)(19,29,31,25)(20,27,21,26)(22,28,32,30);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(18,24)(20,21)(25,29)(28,30);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,25,24,29)(20,28,21,30)(22,26,32,27);(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,26,24,27)(19,28,31,30)(22,25,32,29);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,21,23,20)(18,27,24,26)(19,30,31,28)(22,29,32,25);(1,2)(3,6)(4,5)(7,8)(17,22)(18,28)(19,27)(20,25)(21,29)(23,32)(24,30)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104784);