#8399071

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
prime radical	#35830	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306267);