#8230748

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

5117 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)(17,18,23,27)(19,28,22,24)(20,26,21,25)(29,31,30,32);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32);(9,11)(10,14)(12,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32);(1,4,5,7)(2,3,6,8)(17,22,23,19)(18,24,27,28)(20,29,21,30)(25,32,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263771
maximal normal subgroup of	#72263771
maximal quotient of	#72263792

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8137944);