#8115709

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

5193 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^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, 8022905);