#8126656

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

5015 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:

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

Polycyclic presentation:

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

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, 8033852);