#8873708

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

5372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266767, #72266996, #72266997

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780904);