#8357883

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

4829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8265079);