#8273734

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

5008 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8180930);