#8387598

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

5535 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281380
maximal normal subgroup of	#72281380
maximal quotient of	#72282144, #72282247

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8294794);