#8126653

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	#58991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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, 8033849);