#9490328

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	#45263	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9397524);