#9434676

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

4619 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219279
maximal quotient of	#72219271

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9341872);