#8109956

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

5238 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=gh,e^2=gi,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^b=egi

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, 8017152);