#9017420

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

4779 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16);(3,8)(6,7)(9,10,12,14)(11,13,16,15)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,30)(18,24,27,28)(20,21,26,29)(22,31,32,25);(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,23,26)(18,22,27,32)(19,29,30,21)(24,25,28,31);(1,3)(2,7)(4,6)(5,8)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28);(1,4,5,2)(3,6,8,7)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303333
maximal normal subgroup of	#72303333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924616);