#8702658

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

4884 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8609854);