#8387813

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

5400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^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, 8295009);