#8288178

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

4853 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)(17,18,23,24)(19,26,21,25)(20,29,30,22)(27,32,31,28);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(18,24)(20,30)(25,26)(27,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,25,24,26)(20,27,30,31)(22,32,29,28);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,27)(21,31)(23,30)(24,29)(25,32)(26,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,26,24,25)(20,27,30,31)(22,32,29,28);(1,2)(3,6)(4,5)(7,8)(17,22)(18,20)(19,28)(21,32)(23,29)(24,30)(25,31)(26,27)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8195374);