#8905373

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

5309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219217, #72219279
maximal quotient of	#72219216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8812569);