#8361310

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

5078 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8268506);