#8388169

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

5061 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,22)(21,31)(23,25)(24,32)(26,30)(27,29);(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,15,16,12)(17,19,23,32)(18,24,25,28)(20,29,30,21)(22,31,26,27);(3,7)(6,8)(10,14)(12,15)(18,25)(22,26)(24,28)(27,31);(2,5)(6,8)(10,14)(11,16)(17,20)(18,26)(19,29)(21,32)(22,25)(23,30)(24,27)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,21,23,29)(18,27,25,31)(19,30,32,20)(22,24,26,28);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,15)(14,16)(17,22,23,26)(18,20,25,30)(19,31,32,27)(21,24,29,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8295365);