#8231457

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

5113 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8138653);