#8402302

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

5371 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ghi,e^2=h,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, 8309498);