#8677360

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

4823 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8584556);