#8374266

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

4913 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)(18,23)(24,27)(25,30)(31,32);(1,2,5,4)(3,7,8,6)(11,15)(13,16)(17,18)(19,24)(20,27)(21,25)(22,23)(26,31)(28,32)(29,30);(2,4)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,20)(18,24,23,27)(21,28,29,26)(25,32,30,31);(3,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,27)(21,25)(22,23)(26,31)(28,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,19)(18,24,23,27)(21,28,29,26)(25,31,30,32);(1,4,5,2)(3,7,8,6)(17,21,22,29)(18,25,23,30)(19,26,20,28)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281462);