#8114336

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

5448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021532);