#8701567

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
prime radical	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608763);