#8231415

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

4898 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)(11,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,32)(29,30);(1,2)(3,4)(5,6)(7,8)(10,11)(14,15)(17,19,23,31)(18,24,27,26)(20,29,21,28)(22,30,25,32);(2,6)(4,7)(9,10,13,11)(12,14,16,15)(17,20,23,21)(18,25,27,22)(19,28,31,29)(24,30,26,32);(9,11,13,10)(12,15,16,14)(17,21,23,20)(18,25,27,22)(19,29,31,28)(24,30,26,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,26)(20,28)(21,29)(22,30)(23,31)(24,27)(25,32);(1,4,5,7)(2,3,6,8)(17,22,23,25)(18,20,27,21)(19,30,31,32)(24,29,26,28)

Polycyclic presentation:

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8138611);