#8176351

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

5366 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)(3,7)(4,5)(6,8)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(2,5)(6,7)(10,13)(14,15)(22,30)(25,28)(27,29)(31,32);(9,11)(10,14)(12,16)(13,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=g,f^2=g,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, 8083547);