#8339862

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

5291 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247058);