#8587490

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

5084 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)(5,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,27)(21,24)(22,23)(26,32)(28,29)(30,31);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,22,27)(18,20,23,25)(21,26,30,29)(24,28,31,32);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(18,23)(20,25)(24,31)(28,32);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,32)(26,30)(28,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494686);