#8759259

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

5767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327271, #72336231
maximal normal subgroup of	#72327271, #72336231
maximal quotient of	#72327431, #72327887, #72329350, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8666455);