#8942964

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

5135 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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	#72319739, #72319817
maximal normal subgroup of	#72319739, #72319817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850160);