#8116774

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

4920 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^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, 8023970);