#8182076

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

5069 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:

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

Polycyclic presentation:

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

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, 8089272);