#8509545

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

5199 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)(19,20)(21,26)(22,23)(24,25)(27,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(11,15)(14,16)(18,23)(19,25)(26,31)(27,29);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,23)(20,24)(26,31)(28,32);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,22,25)(18,24,23,20)(21,29,30,27)(26,28,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27);(1,3)(2,6)(4,7)(5,8)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,32)(25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416741);