#8215495

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

4931 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,28)(22,27)(23,25)(29,32)(30,31);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,22)(21,25)(23,28)(26,27)(29,30)(31,32);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(18,25)(19,28)(22,29)(26,31);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,29)(21,27)(22,28)(23,30)(24,32)(25,31);(1,3,4,8)(2,6,5,7)(17,21)(18,19)(20,27)(22,31)(23,24)(25,28)(26,29)(30,32);(1,2)(3,6)(4,5)(7,8)(17,22,23,29)(18,27,25,32)(19,20,28,30)(21,31,24,26)

Polycyclic presentation:

2^9|b^2=g,e^2=ghi,f^2=h,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, 8122691);