#8759258

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

Polycyclic presentation:

2^9|a^2=gh,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	#72327643
maximal normal subgroup of	#72327643
maximal quotient of	#72327923, #72328639, #72329663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8666454);