#8288168

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

4940 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)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(21,29)(24,31)(27,28)(30,32);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32);(1,2)(3,6)(4,5)(7,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,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, 8195364);