#8115877

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
prime radical	#46417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8023073);