#8253943

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

5058 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:

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

Polycyclic presentation:

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

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, 8161139);