#8689759

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

5334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8596955);