#8934915

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

5096 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^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	#72280742
maximal normal subgroup of	#72280742
maximal quotient of	#72280982, #72281224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842111);