#8288388

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

4857 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=hi,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, 8195584);