#8398551

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

5362 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=g,f^2=gi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8305747);