#8395303

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

4912 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)(17,18,23,25)(19,21,20,27)(22,24,30,32)(26,29,28,31);(1,2)(3,8)(4,7)(5,6)(9,11,13,16)(10,14,12,15)(17,18)(19,21)(20,27)(22,24)(23,25)(26,29)(28,31)(30,32);(2,6)(4,8)(10,12)(14,15)(17,19,23,20)(18,21,25,27)(22,28,30,26)(24,31,32,29);(2,6)(3,7)(9,11,13,16)(10,14,12,15)(17,20,23,19)(18,21,25,27)(22,28,30,26)(24,29,32,31);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,20)(19,25)(22,29)(23,27)(24,28)(26,32)(30,31);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,15,16,14)(17,22,23,30)(18,24,25,32)(19,26,20,28)(21,29,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8302499);