#8106070

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

5026 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)(9,10)(11,15)(12,14)(13,16)(18,22)(20,28)(23,26)(27,31);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,16,15)(18,22)(19,25)(24,29)(27,31);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32);(1,3)(2,6)(4,7)(5,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013266);